rewrite the following expression as a single logarithm (using the same base). then evaluate the resulting…

rewrite the following expression as a single logarithm (using the same base). then evaluate the resulting logarithm as a decimal number rounded to four decimal places. $log_{5} 20 + 5log_{5} 4 - log_{5} 5$\nenter the simplified values in the boxes to complete the expression and the evaluation of the logarithm.\nshow hints\nthe expression rewritten as a single logarithm is $log_{square}square$. the result from evaluating the logarithm is $square$.

rewrite the following expression as a single logarithm (using the same base). then evaluate the resulting logarithm as a decimal number rounded to four decimal places. $log_{5} 20 + 5log_{5} 4 - log_{5} 5$\nenter the simplified values in the boxes to complete the expression and the evaluation of the logarithm.\nshow hints\nthe expression rewritten as a single logarithm is $log_{square}square$. the result from evaluating the logarithm is $square$.

Answer

Explanation:

Step1: Apply power rule to $5\log_5 4$

$\log_5 4^5 = \log_5 1024$

Step2: Combine logs using product/quotient rules

$\log_5\left(\frac{20 \times 1024}{5}\right)$

Step3: Simplify the argument

$\frac{20 \times 1024}{5} = 4 \times 1024 = 4096$

Step4: Evaluate $\log_5 4096$

Use change of base: $\log_5 4096 = \frac{\ln 4096}{\ln 5} \approx \frac{8.31776616675}{1.60943791243}$

Answer:

The expression rewritten as a single logarithm is $\log_5 4096$. The result from evaluating the logarithm is $5.1682$.