Math Problem Statement

answer only

Use the Change of Base Formula to rewrite the logarithm with the common logarithm or the natural logarithm: log3(80) = ? Use a calculator to evaluate the logarithm. Round to four decimal places.

Solution

The change of base formula is:

logb(a)=log(a)log(b)orlogb(a)=ln(a)ln(b)\log_b(a) = \frac{\log(a)}{\log(b)} \quad \text{or} \quad \log_b(a) = \frac{\ln(a)}{\ln(b)}

Using log3(80)\log_3(80), we calculate:

log3(80)=log(80)log(3)\log_3(80) = \frac{\log(80)}{\log(3)}

Using a calculator:

log(80)1.9031andlog(3)0.4771\log(80) \approx 1.9031 \quad \text{and} \quad \log(3) \approx 0.4771

log3(80)=1.90310.47713.9893\log_3(80) = \frac{1.9031}{0.4771} \approx 3.9893

Thus, log3(80)3.9893\log_3(80) \approx 3.9893.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Change of Base Formula
Common Logarithms
Natural Logarithms

Formulas

Change of Base Formula: log_b(a) = log(a)/log(b) or log_b(a) = ln(a)/ln(b)

Theorems

Change of Base Theorem

Suitable Grade Level

Grades 10-12