3.06 Logarithm laws and change of base
Write each of the following expressions as a single logarithmic term:
\log_{10} 5 + \log_{10} 4
\log_{10} 18 - \log_{10} 3
\log_{10} 7 - \log_{10} 28
\log_{5} 11 + \log_{5} 2 + \log_{5} 9
\log_{10} 5 + \log_{10} 7 - \log_{10} 3
\log_{7} 12 - \left(\log_{7} 2 + \log_{7} 3\right)
3 \left(\log_{10} 9 + \log_{10} 2\right)
3 \left(\log_{10} 6 - \log_{10} 2\right)
2 \log_{5} 22 - 2 \log_{5} 11
5 \log_{10} 6 + 5 \log_{10} 3
3 + \log_{4} 7
Evaluate the following logarithmic expressions:
\log_{10} 2 + \log_{10} 5
\log_{6} 12 + \log_{6} 18
\log_{2} 72 - \log_{2} 9
\log_{4} 8 + \log_{4} 2
\log_{3} 2 - \log_{3} 18
\log_{6} 12 + \log_{6} 15 - \log_{6} 5
Rewrite the following expressions without any powers or surds:
\log_{4} \left(x^{7}\right)
\log \left(\left(x + 6\right)^{5}\right)
\log \left(\left( 3 x + 7\right)^{ - 1 }\right)
\log \left(x^{\frac{2}{5}}\right)
\log_{10} \sqrt{10}
\log_{10} 10^{\frac{5}{4}}
\log_{10} \left(10^a\right)
\log_{5} \sqrt{x^3}
\log_{5} 125^{\frac{5}{4}}
Rewrite the following as the sum or difference of logarithms without any powers or surds:
\log_{9} \left(u v\right)
\log_{2} \left(5x\right)
\log \left(m^{2}\right)
\log \left(\left( 3 x\right)^{5}\right)
\log \left(\dfrac{1}{x y}\right)
\log \left(\left( 5 x\right)^{ - 7 }\right)
\log \left( 5 x^{\frac{2}{3}}\right)
\log \left(\left( 14 x\right)^{\frac{1}{3}}\right)
Write each of the following as a single logarithm or integer:
5 \log x^{3} - 4 \log x^{2}
5 \log x + 3 \log y
8 \log x - \dfrac{1}{3} \log y
7 \log x - \log \left(\dfrac{1}{x}\right) - \log y
7 \log_{10} 5 - 21 \log_{10} 25
5 \log_{10} 8 - 3 \log_{10} 4
2 \log_{6} 3 + \dfrac{1}{3} \log_{6} 64
\log_{2} 36 - 2 \log_{2} 3
Write \log \left(\dfrac{2 u}{3 v}\right) in terms of \log 2, \log u, \log 3 and \log v.
Rewrite the expression \log x^{2} + \log x^{3} in the form k \log x.
Rewrite the following in terms of \log u and \log v without any powers or surds:
\log \left( u^{3} v^{5}\right)
\log \left(\dfrac{\sqrt[3]{v}}{\sqrt{u}}\right)
Simplify each of the following expressions:
\dfrac{5 \log m^{2}}{6 \log \sqrt[3]{m}}
\dfrac{\log a^{8}}{\log a^{4}}
\dfrac{\log a^{3}}{\log \sqrt[3]{a}}
\dfrac{\log \left(\dfrac{1}{x^{4}}\right)}{\log x}
\dfrac{\log_{10} 4}{\log_{10} 2}
\dfrac{\log_{4} 125}{\log_{4} 5}
\log_{10} 10 + \dfrac{\log_{10} \left(15^{20}\right)}{\log_{10} \left(15^{5}\right)}
\dfrac{8 \log_{10} \left(\sqrt{10}\right)}{\log_{10} \left(100\right)}
If \log_{a} 3 = 1.16 and \log_{a} 2 = 0.73, find the value of \log_{a} \sqrt{54}.
Using the following rounded values, evaluate the expressions below correct to three decimal places:
\log_{10} 7 = 0.845
\log_{10} 2 = 0.301
\log_{10} 3 = 0.477
\log_{10} 25 = 1.398
\log_{10} 5 = 0.699
\log_{10} 28
\log_{10} \left(\dfrac{5}{3}\right)
\log_{10} 49 + \log_{10} 27
\log_{10} 49 - \log_{10} 125
Using the rounded values \log_{x} 3 = 0.62 and \log_{x} 4 = 0.78, find the value of each of the following expressions:
\log_{x} 9
\log_{x} \sqrt{3}
\log_{x} 4 x
\log_{x} \dfrac{1}{3}
\log_{x} 36
Given that \log_{b} x = 2.6 and \log_{b} y = 4.2, find the value of each of the following expressions:
\log_{b} x^{3}
\log_{b} \sqrt[3]{y}
\log_{b} \left( x^{2} \sqrt{y}\right)
\log_{b} \left(\dfrac{b}{x}\right)
Rewrite \log_{3} 20 in terms of base 4 logarithms.
Rewrite the following in terms of base 10 logarithms:
\log_{4} 16
\log_{3} 0.9
\log_{3} \sqrt{5}
Consider the following logarithmic expressions:
Rewrite the expression in terms of base 10 logarithms.
Hence, evaluate each to two decimal places.
If p = \log_c 8 and q = \log_c 10, write the following in terms of p and/or q: