Evaluate the logarithmic expression: log8(1/16)
The Correct Answer and Explanation is:
To evaluate the logarithmic expression log8(116)\log_8 \left(\frac{1}{16}\right)log8(161), we need to understand what the logarithm represents. The general form of a logarithmic expression is:logb(x)=yif and only ifby=x\log_b(x) = y \quad \text{if and only if} \quad b^y = xlogb(x)=yif and only ifby=x
In this case, b=8b = 8b=8, and x=116x = \frac{1}{16}x=161. So, we are looking for the exponent yyy such that:8y=1168^y = \frac{1}{16}8y=161
Step 1: Express 8 and 16 as powers of 2
To make the calculation easier, we can express both 8 and 16 as powers of 2:
- 8=238 = 2^38=23
- 16=2416 = 2^416=24
So, the expression becomes:8y=116=16−1=(24)−1=2−48^y = \frac{1}{16} = 16^{-1} = (2^4)^{-1} = 2^{-4}8y=161=16−1=(24)−1=2−4
Now the equation becomes:(23)y=2−4(2^3)^y = 2^{-4}(23)y=2−4
Step 2: Apply the power rule
Use the power of a power rule (am)n=am⋅n(a^m)^n = a^{m \cdot n}(am)n=am⋅n to simplify the left-hand side:23y=2−42^{3y} = 2^{-4}23y=2−4
Step 3: Solve for yyy
Since the bases on both sides are the same, we can equate the exponents:3y=−43y = -43y=−4
Now, solve for yyy:y=−43y = \frac{-4}{3}y=3−4
Final Answer:
Thus, the value of log8(116)\log_8 \left(\frac{1}{16}\right)log8(161) is −43\frac{-4}{3}3−4.
Explanation:
In this process, we converted both the base (8) and the argument (16) to powers of 2, which allowed us to compare their exponents directly. The final solution shows that to raise 8 to the power of −43\frac{-4}{3}3−4 results in 116\frac{1}{16}161.
