Questions: Evaluate each expression without using a calculator. a. log(1) b. log(10^8) c. 10^(log(4)) d. 10^(8 log(4))

Evaluate each expression without using a calculator. a. log(1) b. log(10^8) c. 10^(log(4)) d. 10^(8 log(4))
Transcript text: Evaluate each expression without using a calculator. a. $\log (1)$ b. $\log \left(10^{8}\right)$ c. $10^{\log (4)}$ d. $10^{8 \log (4)}$
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Solution

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Solution Steps

Step 1: Evaluate \(\log(1)\)

The logarithm of 1 in any base is always 0 because any number raised to the power of 0 is 1. Therefore,

\[ \log(1) = 0 \]

Step 2: Evaluate \(\log\left(10^8\right)\)

Using the property of logarithms \(\log_b(b^x) = x\), we have:

\[ \log\left(10^8\right) = 8 \]

Step 3: Evaluate \(10^{\log(4)}\)

Using the property of logarithms and exponents, \(b^{\log_b(x)} = x\), we have:

\[ 10^{\log(4)} = 4 \]

Final Answer

  • a. \(\boxed{0}\)
  • b. \(\boxed{8}\)
  • c. \(\boxed{4}\)
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