Evaluate the logarithm. Round your answer to the nearest thousandth. log_(5)((1)/(1000))~~◻

Understand the Problem: Understand the problem. We need to find the value of the logarithm of 1/1000 to the base 5.Use Logarithm Properties: Use the logarithm properties. We can use the property of logarithms that states log_b(a/c) = log_b(a) - log_b(c). So, log_(5)((1)/(1000)) = log_(5)(1) - log_(5)(1000).Evaluate log(1): Evaluate log_(5)(1). The logarithm of 1 to any base is 0 because any number to the power of 0 is 1. So, log_(5)(1) = 0.Express 1000 as Power of 5: Express 1000 as a power of 5. We need to express 1000 as 5 to the power of some number. Since 1000 is not a power of 5, we will use the change of base formula. log_(5)(1000) = log(1000) / log(5), where log denotes the common logarithm (base 10).Calculate Log Values: Calculate log(1000) and log(5). Using a calculator, we find: log(1000) = 3 (since 10^3 = 1000) log(5) ≈ 0.69897 (using a calculator).Use Change of Base Formula: Use the change of base formula to find log_(5)(1000). log_(5)(1000) = log(1000) / log(5) = 3 / 0.69897 ≈ 4.29203.Subtract Values: Subtract the values to find the final answer. log_(5)((1)/(1000)) = log_(5)(1) - log_(5)(1000) = 0 - 4.29203 ≈ -4.29203.Round the Answer: Round the answer to the nearest thousandth. Rounded to the nearest thousandth, the value is approximately -4.292.

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_(5)((1)/(1000))~~◻

Evaluate the logarithm. $\newline$ Round your answer to the nearest thousandth. $\newline$ $\log _{5}\left(\frac{1}{1000}\right) \approx \square$

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 _{5}\left(\frac{1}{1000}\right) \approx \square

Understand the Problem: Understand the problem. $\newline$ We need to find the value of the logarithm of $\frac{1}{1000}$ to the base $5$ .
Use Logarithm Properties: Use the logarithm properties. $\newline$ We can use the property of logarithms that states $\log_b\left(\frac{a}{c}\right) = \log_b(a) - \log_b(c)$ . $\newline$ So, $\log_{5}\left(\frac{1}{1000}\right) = \log_{5}(1) - \log_{5}(1000)$ .
Evaluate $\log(1)$ : Evaluate $\log_{5}(1)$ . $\newline$ The logarithm of $1$ to any base is $0$ because any number to the power of $0$ is $1$ . $\newline$ So, $\log_{5}(1) = 0$ .
Express $1000$ as Power of $5$ : Express $1000$ as a power of $5$ . We need to express $1000$ as $5$ to the power of some number. Since $1000$ is not a power of $5$ , we will use the change of base formula. $\log_{5}(1000) = \frac{\log(1000)}{\log(5)}$ , where $\log$ denotes the common logarithm (base $10$ ).
Calculate Log Values: Calculate $\log(1000)$ and $\log(5)$ . Using a calculator, we find: $\log(1000) = 3$ (since $10^3 = 1000$ ) $\log(5) \approx 0.69897$ (using a calculator).
Use Change of Base Formula: Use the change of base formula to find $\log_{5}(1000)$ . $\log_{5}(1000) = \frac{\log(1000)}{\log(5)} = \frac{3}{0.69897} \approx 4.29203$ .
Subtract Values: Subtract the values to find the final answer. $\log_{5}\left(\frac{1}{1000}\right) = \log_{5}(1) - \log_{5}(1000) = 0 - 4.29203 \approx -4.29203$ .
Round the Answer: Round the answer to the nearest thousandth. $\newline$ Rounded to the nearest thousandth, the value is approximately $-4.292$ .