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Express 10 12 in terms of a and b, given 10 2 = a and 10 3 = b.

April 22, 2026Mathematics

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Answer

\mathbf{2a + b}

4 steps

Rewrite $\log_{10} 12$ using prime factorization.
$\log_{10} 12 = \log_{10} (2^2 \times 3)$
Apply the logarithm product rule, $\log (xy) = \log x + \log y$ .
$\log_{10} (2^2 \times 3) = \log_{10} (2^2) + \log_{10} 3$
Apply the logarithm power rule, $\log (x^n) = n \log x$ .
$\log_{10} (2^2) + \log_{10} 3 = 2 \log_{10} 2 + \log_{10} 3$
Substitute the given values, $\log_{10} 2 = a$ and $\log_{10} 3 = b$ .
$2 \log_{10} 2 + \log_{10} 3 = 2a + b$
The expression for $\log_{10} 12$ in terms of $a$ and $b$ is:
$\mathbf{2a + b}$

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