20.05 Logarithms (Extended)
Rewrite each of the following equations in logarithmic form:
5^{2} = 25
3^{x} = 81
3^{1} = 3
2^{0} = 1
4.4^{0} = 1
25^{1.5} = 125
4^{\frac{5}{2}} = 32
4^{ - 2 } = 0.0625
4^{ - 3 } = \dfrac{1}{64}
x^{1.5} = 64
4^{m} = 9
5^{m} = q
w = h^{q}
\dfrac{1}{m^{ - j }} = n
Rewrite each of the following equations in logarithmic form:
10^{2} = 100
10^{ - 2 } = \dfrac{1}{100}
10^{\frac{1}{2}} = \sqrt{10}
10^{ - \frac{1}{2} } = \dfrac{1}{\sqrt{10}}
Rewrite each of the following equations in exponential form:
\log_{4} 16 = 2
\log_{5} 5 = 1
\log_{8} 1 = 0
\log_{2} 0.125 = - 3
\log_{3} \dfrac{1}{3} = - 1
\log_{5.8} 33.64 = 2
\log_{\frac{1}{3}} 9 = - 2
\log_{x} 32= 5
\log_{8} x = 6
\log_{g} 5 = 3
\log_{7} m = 40
\log_{y} k = u
\log_{r} p=y
Rewrite each of the following equations in exponential form:
\log_{10} 10\,000 = 4
\log_{10} \left(\sqrt{10}\right) = \dfrac{1}{2}
\log \left(\dfrac{1}{10\,000}\right) = - 4
\log \left(\dfrac{1}{\sqrt{10}}\right) = -\dfrac{1}{2}
True or false: The equation 4 \left(1.09\right)^{x} = 20 is equivalent to x = \log_{1.09} 5.
For each of the following equations:
Rewrite the equation in logarithmic form.
Find the value of x, correct to two decimal places.
10^{x} = 820
10^{x} = 0.0002
10^{x} = \dfrac{13}{50}
10^{x} = 14\,000
For each of the following numbers:
Express the number as a power of 10.
Hence state the base ten logarithm of the number.
1\,000\,000
0.000\,001
1
\dfrac{1}{10}
Evaluate the following logarithmic expressions without the use of technology:
\log_{2} 64
\log_{2} \left(\dfrac{1}{2}\right)
\log_{5} 0.04
\log_{7} 1
\log_{1.1} 1.21
\log_{0.5} 4
\log_{\frac{1}{5}} 25
\log_{27} 3
\log_{2} \left(\dfrac{1}{16}\right)
Find the value of the following:
The logarithm, with any base, of what number is equal to zero?
In order to estimate \log_{10}90 without a calculator:
Find \log_{10}10.
Find \log_{10}100.
Between which two consecutive integers would you expect \log_{10}90 to lie?
For each of the following common logarithms:
Between which two consecutive integers would you expect the logarithm to lie?
Find the value of the logarithm, correct to four decimal places.
Evaluate the following logarithmic expressions, correct to two decimal places:
\log_{10} 40
\log_{10} 7692
\log_{10} 0.005\,79
\log_{10} 8
\log_{3} 8
\log_{6} 20
\log_{0.5} 7
\log_{0.3} 15
- \log_{2} 23
5 \log_{4} 17
\log_{5} \left( 20^{2} \times \dfrac{3}{43}\right)
\log_{3} 6 + 2 \log_{3} 10 - 5 \log_{3} 15
Find the number such that the base ten logarithm of the number is 2.5. Round your answer to two decimal places.
Find the number such that the base ten logarithm of the number is - 1.2. Round your answer to three decimal places.
Find the number, if the logarithm of the number is 0.