6.01 Definition of the logarithm
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
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
Rewrite each of the following equations in exponential form:
\log_{10} 10\,000 = 4
\log_{10} \left(\sqrt{10}\right) = \dfrac{1}{2}
\log_{10} \left(\dfrac{1}{10\,000}\right) = - 4
\log_{10} \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.
The logarithm of what number is equal to zero?
Evaluate the following logarithmic expressions:
\log_{2} 16
\log_{8} \left(\dfrac{1}{64}\right)
\log_{5} 0.2
\log_{4} 1
\log_{36} 6
\log_{2} \left(\dfrac{1}{4}\right)
\log_{10} 0.1
\log_{7} \sqrt[3]{7}
2^{\log_{2} 3}
\log_{2} \sqrt[4]{2}
\log_{3} 3
\log_{16} \sqrt{2}
\log_{2} \left(\sqrt[3]{\dfrac{1}{16}}\right)
\log_{5} 125^{\frac{5}{4}}
Evaluate \log_{10} 5.16, correct to three decimal places.
In order to estimate \log_{10}90 without a calculator:
Evaluate \log_{10}10.
Evaluate \log_{10}100.
Between which two consecutive integers would you expect \log_{10}90 to lie?
Consider the expression \log 8892.
Between which two consecutive integers would you expect \log 8892 to lie?
Find the value of \log 8892 correct to four decimal places.
Consider the function f \left( x \right) = \log_{10} \left( - 6 x \right). Determine whether each of the following is defined. If so, evaluate the expression correct to two decimal places.
f \left( 3 \right)
f \left( \dfrac{1}{3} \right)
f \left( - \dfrac{1}{3} \right)
f \left( - 3 \right)
Consider the function f \left( x \right) = \log_{2} \left(x - 2\right). Evaluate f \left( 10 \right).
Consider the function f \left( x \right) = \log_{2}x + 3. Evaluate:
f \left( 8 \right)
f \left( \dfrac{1}{8} \right)
f \left( 32 \right)
f \left( \dfrac{1}{64} \right)
True or false: If 5^{x - 6} = 2^{ 4 x + 3}, then x - 6 = \log_{5} 2^{ 4 x + 3}.
Consider the function f \left( x \right) = \log_{10} \left(4 x\right). Solve for the value of x for which f \left( x \right) = 2.
Consider the function f \left( x \right) = \log_{10} \left( 3 x + 9\right). Evaluate f \left( 4 \right) to two decimal places.
Consider the function f \left( x \right) = \log_{4} \left(k x\right) + 1. Solve for the value of k for which f \left( 4 \right) = 3.
Consider the function f \left( x \right) = 4 \log_{\frac{1}{4}} x.
Evaluate f \left( \dfrac{1}{4} \right).
Solve the value for x for which f \left( x \right) = 0.
For each of the following equations:
Rewrite the equation in logarithmic form.
Approximate the value of x to two decimal places.
Given that a > 1, fully simplify the following expressions:
\log_{a} \left(\dfrac{1}{a}\right)
\log_{a} \left(a^{9}\right)
\log_{a} \left(\dfrac{1}{a^{2}}\right)
\log_{a} \left(\sqrt{a}\right)
\log_{a} \left(\dfrac{1}{\sqrt{a}}\right)