Evaluate the logarithm. Round your answer to the nearest thousandth. log_(3)(0.2)~~

Problem Understanding: Understand the problem. We need to find the value of the logarithm of 0.2 with base 3, which is written as log₃(0.2).Evaluation Method: Use a calculator or logarithm properties to evaluate log₃(0.2). Since 0.2 is not a power of 3, we will likely need a calculator to find this logarithm. If a calculator with the ability to compute logarithms with any base is not available, we can use the change of base formula: log₃(0.2) = log(0.2) / log(3) where log denotes the common logarithm (base 10) or natural logarithm (base e).Using Change of Base Formula: Calculate using the change of base formula. Using a scientific calculator, we find: log(0.2) ≈ -0.69897 (using common logarithm) log(3) ≈ 0.47712 (using common logarithm) Now, divide the two values: log₃(0.2) ≈ -0.69897 / 0.47712 ≈ -1.4657Rounding the Result: Round the result to the nearest thousandth. Rounding -1.4657 to the nearest thousandth gives us -1.466.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Evaluate the logarithm.
Round your answer to the nearest thousandth.

log_(3)(0.2)~~

Evaluate the logarithm. $\newline$ Round your answer to the nearest thousandth. $\newline$ $\log _{3}(0.2) \approx$

View step-by-step help

Home

Math Problems

Algebra 1

Compare linear, exponential, and quadratic growth

Full solution

Q. Evaluate the logarithm.

\newline

Round your answer to the nearest thousandth.

\newline

\log _{3}(0.2) \approx

Problem Understanding: Understand the problem. $\newline$ We need to find the value of the logarithm of $0.2$ with base $3$ , which is written as $\log_{3}(0.2)$ .
Evaluation Method: Use a calculator or logarithm properties to evaluate $\log_3(0.2)$ . $\newline$ Since $0.2$ is not a power of $3$ , we will likely need a calculator to find this logarithm. If a calculator with the ability to compute logarithms with any base is not available, we can use the change of base formula: $\newline$ $\log_3(0.2) = \frac{\log(0.2)}{\log(3)}$ $\newline$ where $\log$ denotes the common logarithm (base $10$ ) or natural logarithm (base $e$ ).
Using Change of Base Formula: Calculate using the change of base formula. $\newline$ Using a scientific calculator, we find: $\newline$ $\log(0.2) \approx -0.69897$ (using common logarithm) $\newline$ $\log(3) \approx 0.47712$ (using common logarithm) $\newline$ Now, divide the two values: $\newline$ $\log_3(0.2) \approx \frac{-0.69897}{0.47712} \approx -1.4657$
Rounding the Result: Round the result to the nearest thousandth. $\newline$ Rounding $-1.4657$ to the nearest thousandth gives us $-1.466$ .