Chapter 4.5, Problem 61E

Solution

To find: The error in the given logarithmic expression and correct it.

The error is that $\log 7$ and $\log 3$ are placed incorrectly. Correct expression is ${\log }_{3}7=\frac{\log 7}{\log 3}$.

Given:

An expression ${\log }_{3}7=\frac{\log 3}{\log 7}$.

Formula used:

If $b,m$ and $n$ are positive numbers then ${\log }_{c}a=\frac{{\log }_{b}a}{{\log }_{b}c}$ where $b\ne 1$ and $c\ne 1$.

Calculation:

  $\log 7$ and $\log 3$ are placed incorrectly in the given expression. $\log 7$ must be in the numerator and $\log 3$ must be in the denominator. The correct use of the formula is as follows:

  ${\log }_{3}7$

Apply the formula ${\log }_{c}a=\frac{{\log }_{b}a}{{\log }_{b}c}$.

  ${\log }_{3}7=\frac{\log 7}{\log 3}$

Conclusion:

The error in the expression is that $\log 7$ and $\log 3$ are placed incorrectly. The correct expression is ${\log }_{3}7=\frac{\log 7}{\log 3}$.