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Chapter 33: Problem 960

Given \(\log _{10} 2=0.3010\), find \(\log _{10} 32\)

Short Answer

Expert verified
\( \log_{10} 32 \approx 1.505 \)

Step by step solution

01

Write 32 as a power of 2

Let's express 32 as a power of 2. By recognizing that \(32 = 2^5\), we can write the expression for log as: \[ \log_{10} 32 = \log_{10} 2^5 \]
02

Apply the power rule of logarithms

Now, we will apply the power rule of logarithms which states that for any positive real numbers \(a\) and \(b\), and any integer \(n\), we have: \[ \log_{a}(b^n) = n \log_{a}(b) \] Applying the rule to our expression, we get: \[ \log_{10} 32 = \log_{10} 2^5 = 5\log_{10} 2 \]
03

Substitute the known value

We are given the value of \(\log_{10} 2 = 0.3010\). Now, we will substitute this value into the expression and solve for \(\log_{10} 32\): \[ \log_{10} 32 = 5\log_{10} 2 = 5(0.3010) \]
04

Perform the multiplication

Now we just need to multiply 5 by 0.3010: \[ 5(0.3010) = 1.505 \] So, \(\log_{10} 32 \approx 1.505\).

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