Finding the Logarithm Base: We want to find the value of log_(8)2. This means we are looking for the exponent that 8 must be raised to in order to get 2.Expressing 8 as a Power of 2: We can express 8 as a power of 2, since 8 is 2 cubed (2^3). This will help us to simplify the logarithm.Rewriting the Logarithm: Now we rewrite the logarithm using the fact that 8 is 2^3: log_(2^3)2.Simplifying Using Logarithm Property: Using the property of logarithms that log_(a^b)c = (1/b) * log_a(c), we can simplify our expression to 1/3 * log_(2)2.Determining the Value of log(2)2: We know that log_(2)2 is 1, because 2 raised to the power of 1 is 2.Calculating the Final Answer: Multiplying 1/3 by 1 gives us the final answer: 1/3 * 1 = 1/3.

Grade 8

Relationship between squares and square roots

Full solution

\log _{8} 2=

Finding the Logarithm Base: We want to find the value of $\log_{8}2$ . This means we are looking for the exponent that $8$ must be raised to in order to get $2$ .
Expressing $8$ as a Power of $2$ : We can express $8$ as a power of $2$ , since $8$ is $2$ cubed ( $2^3$ ). This will help us to simplify the logarithm.
Rewriting the Logarithm: Now we rewrite the logarithm using the fact that $8$ is $2^3$ : $\log_{(2^3)} 2$ .
Simplifying Using Logarithm Property: Using the property of logarithms that $\log_{a^b}c = \frac{1}{b} \cdot \log_a(c)$ , we can simplify our expression to $\frac{1}{3} \cdot \log_{2}2$ .
Determining the Value of $\log(2)2$ : We know that $\log_{2}2$ is $1$ , because $2$ raised to the power of $1$ is $2$ .
Calculating the Final Answer: Multiplying $\frac{1}{3}$ by $1$ gives us the final answer: $\frac{1}{3} \times 1 = \frac{1}{3}$ .