Question

use the change of base formula to rewrite the logarithm with the common logarithm or the natural logarithm:
\\(\log_{5}(59)\\)
\\(\log_{5}(59) = \\)

use a calculator to evaluate the logarithm. round to four decimal places.

question help: \\(\boldsymbol{\text{message instructor}}\\)
\\(\boldsymbol{> \text{next question}}\\)

Explanation:

(Rewriting):

Step1: Recall Change of Base Formula

The Change of Base Formula states that for any positive numbers \( a \), \( b \), and \( c \) (where \( a
eq 1 \) and \( c
eq 1 \)), \( \log_{a}(b)=\frac{\log_{c}(b)}{\log_{c}(a)} \).

Step2: Apply the Formula to \( \log_{5}(59) \)

We can use either the common logarithm (base 10) or the natural logarithm (base \( e \)) for \( c \). Using the common logarithm (base 10), we substitute \( a = 5 \), \( b = 59 \), and \( c = 10 \) into the formula. So, \( \log_{5}(59)=\frac{\log(59)}{\log(5)} \). If we use the natural logarithm, it would be \( \frac{\ln(59)}{\ln(5)} \).

(Evaluating):

Step1: Calculate the Numerator and Denominator

First, calculate \( \log(59) \approx 1.770976 \) (using a calculator for base 10 logarithm) and \( \log(5) \approx 0.698970 \).

Step2: Divide the Two Values

Now, divide the two results: \( \frac{1.770976}{0.698970}\approx 2.5464 \) (rounded to four decimal places). If we used the natural logarithm, \( \ln(59)\approx 4.077537 \) and \( \ln(5)\approx 1.609438 \), and \( \frac{4.077537}{1.609438}\approx 2.5464 \) as well.

Answer:

(Rewriting):
$\frac{\log(59)}{\log(5)}$ (or $\frac{\ln(59)}{\ln(5)}$)