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
Write the following equations in logarithmic form.
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.
Consider the functions f \left( x \right) = \log_{2} x + \log_{2} \left( 3 x - 4\right) and g \left( x \right) = \log_{2} \left( 4 x - 4\right).
Evaluate f \left( 2 \right).
Evaluate g \left( 2 \right).
Is f \left( x \right) = g \left( x \right)?
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_{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}x + 3. Evaluate:
f \left( 8 \right)
f \left( \dfrac{1}{8} \right)
f \left( 32 \right)
f \left( \dfrac{1}{64} \right)
Suppose f \left( x \right) = \log_{a} x and f\left(2\right) = 3. Find:
f\left(4\right)
f\left(\dfrac{1}{8}\right)
f^{ - 1 }\left(0\right)
f^{ - 1 }\left( - 3 \right)
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_{4} \left(k x\right) + 1. Solve for the value of k for which f \left( 4 \right) = 3.
After an earthquake in their home town, a family started a fund which would help cover the basic needs of those affected. People donated to the fund, and the value of the fund grew according to t = 25 \log_{14} \left(A + 1\right), where A is the amount raised (in thousands of dollars) after t days.
The family had a certain target in mind, and wanted to know how many days it would take to reach the target. They substituted A = 5000 into the formula and got the result t = 81. What do the values 5000 and 81 represent?
Find the value of 5 \log_{e} e.
Is the value of \log_{e} 2 greater than or less than 1?
Consider the following logarithmic expressions:
\log_{e} 7, \log_{3} 7, \log_{2} 7Which expression has the largest value? Explain your answer.
Evaluate \log 23 writing the answer to the nearest thousandth.
Consider x=\ln 31. Find the value of x, correct to two decimal places.
Evaluate each of the following expressions:
\ln e^{3.5}
\ln e^{4}
\sqrt{6} \ln \left(e^{\sqrt{6}}\right)
\ln \left(\dfrac{1}{e^{2}}\right)
Find the exact value of x in each of the following:
3 \ln x = 9
\ln 3 x = 5
2 \ln 3 x = 6
5 e^{x} = 25
4 e^{x + 7} = 20
\ln \left( 2 x - 1\right) = \ln \left(x + 3\right)
\ln x + 2 = \ln \left( 2 x + 1\right)
e^{ - 5 \ln x } = \dfrac{1}{243}
\ln e^{x} - 3 \ln e = \ln e^{2}
e^{x + \ln 8} = 5 e^{x} + 3
\ln e^{\ln \left(x - 1\right)} - \ln \left(x - 7\right) = \ln 4
e^{ 2 x} - 8 e^{x} + 7 = 0
e^{ 2 x} - 7 e^{x} + 12 = 0
3 e^{ 2 x} - 2 e^{x} = 8
\dfrac{1}{3} e^{ 2 x} + 2 e^{x} = 9
\ln^{2} x - 4 \ln x = 5