Identify base and argument: Identify the base of the logarithm and the argument. The base of the logarithm is 27, and the argument is 1/3.Recognize logarithm of 1: Recognize that any logarithm of 1 is 0, regardless of the base. log_b(1) = 0 for any base b. Since the argument is 1/3, we need to find a way to relate 1/3 to the base 27.Express base as power of 3: Express the base 27 as a power of 3. Since 27 is a power of 3, we can write it as 27 = 3^3.Rewrite logarithm with new base: Rewrite the logarithm with the new base expression. log_(27)(1/3) can be rewritten as log_(3^3)(1/3).Apply change of base formula: Apply the change of base formula for logarithms. The change of base formula is log_b(a) = log_c(a) / log_c(b). Using this formula, we can rewrite the expression as log_(3)(1/3) / log_(3)(3^3).Evaluate logarithms: Evaluate the logarithms. log_(3)(1/3) is asking ＂3 to what power gives 1/3?＂, which is -1 because 3^(-1) = 1/3. log_(3)(3^3) is asking ＂3 to what power gives 3^3?＂, which is 3 because 3^3 = 27. So we have (-1) / 3.Perform division for final answer: Perform the division to get the final answer. (-1) / 3 equals -1/3.

Resources

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$\log _{27} \frac{1}{3}=$

View step-by-step help

Home

Math Problems

Algebra 2

Evaluate logarithms using properties

Full solution

\log _{27} \frac{1}{3}=

Identify base and argument: Identify the base of the logarithm and the argument. $\newline$ The base of the logarithm is $27$ , and the argument is $\frac{1}{3}$ .
Recognize logarithm of $1$ : Recognize that any logarithm of $1$ is $0$ , regardless of the base. $\newline$ $\log_b(1) = 0$ for any base $b$ . $\newline$ Since the argument is $\frac{1}{3}$ , we need to find a way to relate $\frac{1}{3}$ to the base $27$ .
Express base as power of $3$ : Express the base $27$ as a power of $3$ . $\newline$ Since $27$ is a power of $3$ , we can write it as $27 = 3^3$ .
Rewrite logarithm with new base: Rewrite the logarithm with the new base expression. $\log_{27}(\frac{1}{3})$ can be rewritten as $\log_{3^3}(\frac{1}{3})$ .
Apply change of base formula: Apply the change of base formula for logarithms. $\newline$ The change of base formula is $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ . $\newline$ Using this formula, we can rewrite the expression as $\frac{\log_{3}(\frac{1}{3})}{\log_{3}(3^3)}$ .
Evaluate logarithms: Evaluate the logarithms. $\newline$ $\log_{3}(\frac{1}{3})$ is asking ＂ $3$ to what power gives $\frac{1}{3}$ ?＂, which is $-1$ because $3^{-1} = \frac{1}{3}$ . $\newline$ $\log_{3}(3^3)$ is asking ＂ $3$ to what power gives $3^3$ ?＂, which is $3$ because $3^3 = 27$ . $\newline$ So we have $(-1) / 3$ .
Perform division for final answer: Perform the division to get the final answer. $\newline$ $(-1) / 3$ equals $-\frac{1}{3}$ .