Question

use properties of logarithms to expand each logarithmic expression as much as possible. evaluate logarithmic expressions without using a calculator if possible. \\(\log_{10}(10x)=\\) \\(\log_{10}(10x)=\square\\) this test: 25 point(s) possible this question: 1 point(s) possible

Explanation:

Step1: Recall log product rule

The product rule of logarithms states that \(\log_b(MN)=\log_b(M)+\log_b(N)\). For \(\log_{10}(10x)\), we can apply this rule where \(M = 10\) and \(N=x\), so \(\log_{10}(10x)=\log_{10}(10)+\log_{10}(x)\).

Step2: Evaluate \(\log_{10}(10)\)

We know that for a logarithm with base \(b\), \(\log_b(b) = 1\) (by the definition of a logarithm: if \(y=\log_b(b)\), then \(b^y=b\), so \(y = 1\)). Here, \(b = 10\), so \(\log_{10}(10)=1\).

Step3: Substitute back

Substituting \(\log_{10}(10)=1\) into the expression from Step 1, we get \(\log_{10}(10x)=1+\log_{10}(x)\).

Answer:

\(1 + \log_{10}(x)\)