Using the Properties of Logarithms

Examples

_{Rewrite each expression. Assume all variables represent positive real numbers, with $a\ne1$ and $b\ne1.$}

$LaTeX: \log_6\left(7\cdot9\right)$ $\log_6\left(7\cdot9\right)$
$LaTeX: \log_9\frac{15}{7}$ $\log_9\frac{15}{7}$
$LaTeX: \log_3\sqrt[]{8}$ $\log_3\sqrt[]{8}$
$LaTeX: \log_a\sqrt[3]{m^2}$ $\log_a\sqrt[3]{m^2}$
$LaTeX: \log_a\frac{mnq}{p^2t^4}$ $\log_a\frac{mnq}{p^2t^4}$
$LaTeX: \log_b\sqrt[n]{\frac{x^3y^5}{z^m}}$ $\log_b\sqrt[n]{\frac{x^3y^5}{z^m}}$

Solutions

$LaTeX: \log_6\left(7\cdot9\right)=\log_67+\log_69$ $\log_6\left(7\cdot9\right)=\log_67+\log_69$
$LaTeX: \log_9\frac{15}{7}=\log_915-\log_97$ $\log_9\frac{15}{7}=\log_915-\log_97$
$LaTeX: \log_3\sqrt[]{8}=\log_3\left(8^{\frac{1}{2}}\right)=\frac{1}{2}\log_38$ $\log_3\sqrt[]{8}=\log_3\left(8^{\frac{1}{2}}\right)=\frac{1}{2}\log_38$
$LaTeX: \log_a\sqrt[3]{m^2}=\log_am^{\frac{2}{3}}=\frac{2}{3}\log_am$ $\log_a\sqrt[3]{m^2}=\log_am^{\frac{2}{3}}=\frac{2}{3}\log_am$
$LaTeX: \log_a\frac{mnq}{p^2t^4}=\log_a\left(mnq\right)-\log_a\left(p^2t^4\right)=\log_am+\log_an+\log_aq-\left(\log_ap^2+\log_at^4\right)$ $\log_a\frac{mnq}{p^2t^4}=\log_a\left(mnq\right)-\log_a\left(p^2t^4\right)=\log_am+\log_an+\log_aq-\left(\log_ap^2+\log_at^4\right)$
$LaTeX: =\log_am+\log_an+\log_aq-\left(2\log_ap+4\log_at\right)=\log_am+\log_an+\log_aq-2\log_ap-4\log_at$ $=\log_am+\log_an+\log_aq-\left(2\log_ap+4\log_at\right)=\log_am+\log_an+\log_aq-2\log_ap-4\log_at$
$LaTeX: \log_b\sqrt[n]{\frac{x^3y^5}{z^m}}=\log_b\left(\frac{x^3y^5}{z^m}\right)^{\frac{1}{n}}=\frac{1}{n}\log_b\left(\frac{x^3y^5}{z^m}\right)$ $\log_b\sqrt[n]{\frac{x^3y^5}{z^m}}=\log_b\left(\frac{x^3y^5}{z^m}\right)^{\frac{1}{n}}=\frac{1}{n}\log_b\left(\frac{x^3y^5}{z^m}\right)$
$LaTeX: =\frac{1}{n}\left(\log_bx^3+\log_by^5-\log_bz^m\right)=\frac{1}{n}\left(3\log_bx+5\log_by-mlog_bz\right)$ $=\frac{1}{n}\left(\log_bx^3+\log_by^5-\log_bz^m\right)=\frac{1}{n}\left(3\log_bx+5\log_by-mlog_bz\right)$
$LaTeX: =\frac{3}{n}\log_bx+\frac{5}{n}\log_by-\frac{m}{n}\log_bz$ $=\frac{3}{n}\log_bx+\frac{5}{n}\log_by-\frac{m}{n}\log_bz$

Examples

Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with $LaTeX: a\ne1$ $a\ne1$ and $LaTeX: b\ne1.$ $b\ne1.$

$LaTeX: \log_3\left(x+2\right)+\log_3x-\log_32$ $\log_3\left(x+2\right)+\log_3x-\log_32$
$LaTeX: 2\log_am-3\log_an$ $2\log_am-3\log_an$
$LaTeX: \frac{1}{2}\log_bm+\frac{3}{2}\log_b2n-\log_bm^2n$ $\frac{1}{2}\log_bm+\frac{3}{2}\log_b2n-\log_bm^2n$

Solutions

$LaTeX: \log_3\left(x+2\right)+\log_3x-\log_32=\log_3\frac{\left(x+2\right)x}{2}$ $\log_3\left(x+2\right)+\log_3x-\log_32=\log_3\frac{\left(x+2\right)x}{2}$
$LaTeX: 2\log_am-3\log_an=\log_am^2-\log_an^3=\log_a\frac{m^2}{n^3}$ $2\log_am-3\log_an=\log_am^2-\log_an^3=\log_a\frac{m^2}{n^3}$
$LaTeX: \frac{1}{2}\log_bm+\frac{3}{2}\log_b2n-\log_bm^2n=\log_bm^{\frac{1}{2}}+\log_b\left(2n\right)^{\frac{3}{2}}-\log_bm^2n$ $\frac{1}{2}\log_bm+\frac{3}{2}\log_b2n-\log_bm^2n=\log_bm^{\frac{1}{2}}+\log_b\left(2n\right)^{\frac{3}{2}}-\log_bm^2n$

$LaTeX: =\log_b\frac{m^{\frac{1}{2}}\left(2n\right)^{\frac{3}{2}}}{m^2n}=\log_b\frac{2^{\frac{3}{2}}n^{\frac{1}{2}}}{m^{\frac{3}{2}}}=\log_b\left(\frac{2^3n}{m^3}\right)^{\frac{1}{2}}=\log_b\sqrt[]{\frac{8n}{m^3}}$ $=\log_b\frac{m^{\frac{1}{2}}\left(2n\right)^{\frac{3}{2}}}{m^2n}=\log_b\frac{2^{\frac{3}{2}}n^{\frac{1}{2}}}{m^{\frac{3}{2}}}=\log_b\left(\frac{2^3n}{m^3}\right)^{\frac{1}{2}}=\log_b\sqrt[]{\frac{8n}{m^3}}$

Caution:

There is no property of logarithms to rewrite a logarithm of a sum or difference. That is why log3(x + 2) was not written as log3 x + log3 2. The distributive property does not apply in a situation like this because log3 (x + y) is one term. The abbreviation “log” is a function name, not a factor.

Examples Using the Properties of Logarithms with Numerical Values

Assume that $LaTeX: \log_{10}2\approx0.3010,\:$ $\log_{10}2\approx0.3010,\:$ find each logarithm without using a scientific calculator.

$LaTeX: \log_{10}4$ $\log_{10}4$
$LaTeX: \log_{10}5$ $\log_{10}5$

Solutions

$LaTeX: \log_{10}4=\log_{10}2^2=2\log_{10}2\approx2\left(0.3010\right)\approx0.6020$ $\log_{10}4=\log_{10}2^2=2\log_{10}2\approx2\left(0.3010\right)\approx0.6020$
$LaTeX: \log_{10}5=\log_{10}\frac{10}{2}=\log_{10}10-\log_{10}2\approx1-0.3010\approx0.6990$ $\log_{10}5=\log_{10}\frac{10}{2}=\log_{10}10-\log_{10}2\approx1-0.3010\approx0.6990$