Chapter 4.4, Problem 72E

a.

To determine

Whether the equation or expression “ $\log \left(\frac{x}{y}\right)=\frac{\log x}{\log y}$ .” is true or false.

Answer to Problem 72E

The expression “ $\log \left(\frac{x}{y}\right)=\frac{\log x}{\log y}$ .” is false.

Explanation of Solution

Given information:

The expression “ $\log \left(\frac{x}{y}\right)=\frac{\log x}{\log y}$ .”

Consider the provided statement “ $\log \left(\frac{x}{y}\right)=\frac{\log x}{\log y}$ .”

The statement is false.

The logarithmic function is a quantity representing the power to which a fixed number such as base must be raised to a produce a given number.

Recall that the function is written mathematically, it is represented as when two values are divided with the same base is to subtract the exponents.

Therefore, the rule for division is to subtract the logarithms, $\log \left(\frac{x}{y}\right)=\log x-\log y$ .

Now it is clear that the function is $\log \left(\frac{x}{y}\right)=\frac{\log x}{\log y}$ false.

In mathematics, the logarithm is the inverse function to exponent to which another fixed number $x$ is the exponent to which another fixed number be raised, to produce that number $x$ .

For example: Consider a expression, $\log \left(\frac{3}{4}\right)$ .

Observe the function is solved by the identity $\log \left(\frac{x}{y}\right)=\log x-\log y$ .

Mathematically, it is written as:

  $\log \left(\frac{3}{4}\right)=\log 3-\log 4$

Therefore, the function is simplified easily with the division identity $\log \left(\frac{x}{y}\right)=\log x-\log y$ .

Thus, the statement “ $\log \left(\frac{x}{y}\right)=\frac{\log x}{\log y}$ .” is false.

b.

To determine

Whether the equation or expression “ ${\log }_2\left(x-y\right)={\log }_{2}x-{\log }_{2}y$ .” is true or false.

Answer to Problem 72E

The expression “ ${\log }_2\left(x-y\right)={\log }_{2}x-{\log }_{2}y$ .” is false.

Explanation of Solution

Given information:

The expression “ ${\log }_2\left(x-y\right)={\log }_{2}x-{\log }_{2}y$ .”

Consider the provided statement “ ${\log }_2\left(x-y\right)={\log }_{2}x-{\log }_{2}y$ .”

The statement is false.

The log of a difference is NOT the difference of the logs. The difference of the logs is the log of the quotient.

So the log of the difference cannot be simplified as ${\log }_2\left(x-y\right)\ne {\log }_{2}x-{\log }_{2}y$ .

Mathematically The logarithmic function is a quantity representing the power to which a fixed number such as base must be raised to a produce a given number.

Recall that the function is written mathematically, it is represented as when two values are divided with the same base is to subtract the exponents.

Therefore, the rule for difference is to subtract the logarithms, ${\log }_2\left(x-y\right)=\frac{\log \left(x-y\right)}{\log 2}$ .

Now it is clear that the function is ${\log }_2\left(x-y\right)=\frac{\log \left(x-y\right)}{\log 2}$ false.

In mathematics, the logarithm is the inverse function to exponent to which another fixed number $x$ is the exponent to which another fixed number be raised, to produce that number $x$ .

For example: Consider a expression, ${\log }_2\left(3-4\right)=\frac{\log \left(3-4\right)}{\log 2}$ .

Thus, the statement “ ${\log }_2\left(x-y\right)=\frac{\log \left(x-y\right)}{\log 2}$ .” is false.

c.

To determine

Whether the equation or expression “ ${\log }_5\left(\frac{a}{b^2}\right)={\log }_{5}a-2{\log }_{5}b$ .” is true or false.

Answer to Problem 72E

The expression “ ${\log }_5\left(\frac{a}{b^2}\right)={\log }_{5}a-2{\log }_{5}b$ .” is true.

Explanation of Solution

Given information:

The expression “ ${\log }_5\left(\frac{a}{b^2}\right)={\log }_{5}a-2{\log }_{5}b$ .”

Consider the provided statement “ ${\log }_5\left(\frac{a}{b^2}\right)={\log }_{5}a-2{\log }_{5}b$ .”

The statement is true

Mathematically The logarithmic function is a quantity representing the power to which a fixed number such as base must be raised to a produce a given number.

Recall that the function is written mathematically, it is represented as when two values are divided with the same base is to subtract the exponents.

Therefore, the base rule for difference is to divide the logarithms with base, ${\log }_5\left(\frac{a}{b^2}\right)={\log }_{5}a-2{\log }_{5}b$ .

In mathematics, the logarithm is the inverse function to exponent to which another fixed number $x$ is the exponent to which another fixed number be raised, to produce that number $x$ .

For example: Consider a expression, ${\log }_5\left(\frac{3}{4^2}\right)={\log }_{5}3-2{\log }_{5}4$ .

Thus, the statement “ ${\log }_5\left(\frac{a}{b^2}\right)={\log }_{5}a-2{\log }_{5}b$ .” is true.

d.

To determine

Whether the equation or expression “ $\log 2^z=z\log 2$ .” is true or false.

Answer to Problem 72E

The expression “ $\log 2^z=z\log 2$ .” is true.

Explanation of Solution

Given information:

The expression “ $\log 2^z=z\log 2$ .”

Consider the provided statement “ $\log 2^z=z\log 2$ .”

The statement is true.

The power of log function multiplies with the function and the rest of the function written as same.

Mathematically The logarithmic function is a quantity representing the power to which a fixed number such as base must be raised to a produce a given number.

Recall that the function is written mathematically, it is represented as when the number has power as exponent of the function then the power comes forward and multiply with the function.

Therefore, the base rule for logarithms with base and exponent function is, $\log 2^z=z\log 2$ .

In mathematics, the logarithm is the inverse function to exponent to which another fixed number $x$ is the exponent to which another fixed number be raised, to produce that number $x$ .

For example: Consider a expression, ${\log }_5{x}^3$ .

Rewrite the function as: $\log x^3=3\log x$

Thus, the statement “ $\log 2^z=z\log 2$ .” is true.

e.

To determine

Whether the equation or expression “ $\left(\log p\right)\left(\log q\right)=\log p+\log q$ .” is true or false.

Answer to Problem 72E

The expression “ $\left(\log p\right)\left(\log q\right)=\log p+\log q$ .” is false.

Explanation of Solution

Given information:

The expression “ $\left(\log p\right)\left(\log q\right)=\log p+\log q$ .”

Consider the provided statement “ $\left(\log p\right)\left(\log q\right)=\log p+\log q$ .”

The statement is false.

The multiplication rule of log is when two values with the same base together, the rule is that keep the base same and add the exponents. Well logarithms are exponents and when for multiplication we did add the logarithms.

So the given function is not the correct function $\left(\log p\right)\left(\log q\right)\ne \log p+\log q$ .

This function is exactly equal to $\left(\log p\right)\left(\log q\right)=\log \left(p\right)\log \left(q\right)$

For example: Consider a expression, $\left(\log 3\right)\left(\log 4\right)$ .

Rewrite the function as: $\left(\log 3\right)\left(\log 4\right)=\log \left(3\right)\log \left(4\right)$

Thus, the statement “ $\left(\log p\right)\left(\log q\right)=\log p+\log q$ .” is false.

f.

To determine

Whether the equation or expression “ $\frac{\log a}{\log b}=\log a-\log b$ .” is true or false.

Answer to Problem 72E

The expression “ $\frac{\log a}{\log b}=\log a-\log b$ .” is true.

Explanation of Solution

Given information:

The expression “ $\frac{\log a}{\log b}=\log a-\log b$ .”

Consider the provided statement “ $\frac{\log a}{\log b}=\log a-\log b$ .”

The statement is true

Mathematically The logarithmic function is a quantity representing the power to which a fixed number such as base must be raised to a produce a given number.

Recall that the function is written mathematically, it is represented as when two values are divided with the same base is to subtract the exponents.

Therefore, the base rule for divide the logarithms with base, $\frac{\log a}{\log b}=\log a-\log b$ .

In mathematics, the logarithm is the inverse function to exponent to which another fixed number $x$ is the exponent to which another fixed number be raised, to produce that number $x$ .

For example: Consider a expression, $\frac{\log 3x}{\log 4x}=\log 3x-\log 4x$ .

Thus, the statement “ $\frac{\log a}{\log b}=\log a-\log b$ .” is true.

g.

To determine

Whether the equation or expression “ ${\left({\log }_{2}7\right)}^x=x{\log }_{2}7$ .” is true or false.

Answer to Problem 72E

The expression “ ${\left({\log }_{2}7\right)}^x=x{\log }_{2}7$ .” is false.

Explanation of Solution

Given information:

The expression “ ${\left({\log }_{2}7\right)}^x=x{\log }_{2}7$ .”

Consider the provided statement “ ${\left({\log }_{2}7\right)}^x=x{\log }_{2}7$ .”

The statement is false.

The multiplication rule of log is when two values with the same base together, the rule is that keep the base same and add the exponents. Well logarithms are exponents and when for multiplication we did add the logarithms.

Here when log base function are given and whole power is also given then the power comes forward and the rest of the function as same.

So the given function is not the correct function ${\left({\log }_{2}7\right)}^x\ne x{\log }_{2}7$ .

This function is exactly equal to:

  $\begin{array}{c}{\left({\log }_{2}7\right)}^x=x{\log }_{2}7\\ =\frac{\log {\left(7\right)}^x}{\log {\left(2\right)}^x}\end{array}$

For example: Consider a expression, ${\left({\log }_{2}3\right)}^x$ .

Rewrite the function as:

  ${\left({\log }_{2}3\right)}^x=\frac{\log {\left(3\right)}^x}{\log {\left(2\right)}^x}$

Thus, the statement “ ${\left({\log }_{2}7\right)}^x=x{\log }_{2}7$ .” is false.

c.

To determine

Whether the equation or expression “ ${\log }_a{a}^a=a$ .” is true or false.

Answer to Problem 72E

The expression “ ${\log }_a{a}^a=a$ .” is true.

Explanation of Solution

Given information:

The expression “ ${\log }_a{a}^a=a$ .”

Consider the provided statement “ ${\log }_a{a}^a=a$ .”

The statement is true

Mathematically The logarithmic function is a quantity representing the power to which a fixed number such as base must be raised to a produce a given number.

Recall that the function is written mathematically, it is represented as when ${\log }_a$ and the exponent also has power $a^a$ then according to the logarithm rules ${\log }_{a}a=1$ .

Therefore, the base rule logarithms with base, ${\log }_a{a}^a=a$ .

In mathematics, the logarithm is the inverse function to exponent to which another fixed number $x$ is the exponent to which another fixed number be raised, to produce that number $x$ .

For example: Consider a expression, ${\log }_x{x}^x=x{\log }_{x}x$ .

Observe that the expression has ${\log }_{x}x$ that is equal to 1then the equation becomes:

  $\begin{array}{c}{\log }_x{x}^x=x{\log }_{x}x\\ =x\end{array}$

Thus, the statement “ ${\log }_a{a}^a=a$ .” is true.

i.

To determine

Whether the equation or expression “ $\log \left(x-y\right)=\frac{\log x}{\log y}$ .” is true or false.

Answer to Problem 72E

The expression “ $\log \left(x-y\right)=\frac{\log x}{\log y}$ .” is true.

Explanation of Solution

Given information:

The expression “ $\log \left(x-y\right)=\frac{\log x}{\log y}$ .”

Consider the provided statement “ $\log \left(x-y\right)=\frac{\log x}{\log y}$ .”

The statement is true.

Mathematically The logarithmic function is a quantity representing the power to which a fixed number such as base must be raised to a produce a given number.

Recall that the function is written mathematically, it is represent that the difference of two logarithm functions are written as $\log \left(x-y\right)=\frac{\log x}{\log y}$ .

Mathematically,

  $\begin{array}{l}\log \left(x-y\right)=0\\ \log \left(x-y\right)=\frac{\log x}{\log y}\end{array}$

So the given expression is true.

In mathematics, the logarithm is the inverse function to exponent to which another fixed number $x$ is the exponent to which another fixed number be raised, to produce that number $x$ .

For example: Consider a expression,

  $\log \left(3x-4y\right)=\frac{\log 3x}{\log 4y}$ .

Thus, the statement “ $\log \left(x-y\right)=\frac{\log x}{\log y}$ .” is true.

i.

To determine

Whether the equation or expression “ $-\ln \left(\frac{1}{A}\right)=\ln A$ .” is true or false.

Answer to Problem 72E

The expression “ $-\ln \left(\frac{1}{A}\right)=\ln A$ .” is true.

Explanation of Solution

Given information:

The expression “ $-\ln \left(\frac{1}{A}\right)=\ln A$ .”

Consider the provided statement “ $-\ln \left(\frac{1}{A}\right)=\ln A$ .”

The statement is true.

Mathematically The logarithmic function is a quantity representing the power to which a fixed number such as base must be raised to a produce a given number.

The natural logarithm of a number is its logarithm to base of the mathematical constant $e$ , where $e$ is an irrational and transcendental number.

The natural logarithm of $x$ is generally written as $\ln x,{\log }_{e}x$ or sometimes, if the base $e$ is implicit, simply $\log x$ .

Recall that the function is written mathematically, it is represent that the difference of two the logarithmic functions are written as $-\ln \left(\frac{1}{A}\right)=\ln A$ .

The given function an, $-\ln \left(\frac{1}{A}\right)=\ln A$ logarithmic identity, So the given expression is true.

In mathematics, the logarithm is the inverse function to exponent to which another fixed number $x$ is the exponent to which another fixed number be raised, to produce that number $x$ .

For example: Consider a expression,

  $-\ln \left(\frac{1}{5}\right)=\ln 5$ .

Thus, the statement “ $-\ln \left(\frac{1}{A}\right)=\ln A$ .” is true.