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Subsection 2.8 Properties of Logarithms Exercises
Rewrite the expressions using a single natural logarithm.
\(\displaystyle \ln a + 4\ln b - 3\ln c\)
\(\displaystyle \frac{1}{2}\ln s - 2\ln t - \frac{1}{2}\ln c\)
Solve for
\(x\) in the following equation:
\(\ln (x-2) + \ln (x-5) = \ln 4\text{.}\)
Solve for
\(t\) in the following equation:
\(3^{t-1}7^{t-2}=27783\text{.}\)
Solve
\(e^{t^2-2t-10}=200\text{.}\)
Solve. Give your answer in exact form.
\(\displaystyle 2\ln(x-2) - \ln x = 1\)
\(\displaystyle \log_{10}(x-3) + \log_{10}(x+5) = 1 + \log_{10} 2\)
\(\displaystyle e^{2x} - 3e^x = 10\)
\(\displaystyle 7^{2t} + 4\cdot 7^t - 32 = 0\)
Recall that any object experiencing the force of gravity can be modeled by the equation
\(h(t)=-16t^2+vt + c\text{,}\) where
\(t\) is the time in seconds,
\(h(t)\) is the height in feet,
\(c\) is the initial height of the object, and
\(v\) is the initial velocity of the object. A soccer ball is kicked from a height of 2 feet above the ground and has an initial velocity of 70 feet per second.
What is the highest the ball will go?
During what time interval is the ball at least 10 feet in the air?
How long is the ball in the air?
Let
\(f(x)=2e^x\text{,}\) \(g(x)=2x-1\) and
\(k(x)=x^2\text{.}\)
Evaluate
\(f(g(-1))\text{.}\)
Write an expression for
\((k \circ g)(x)\text{.}\)
Write an expression for
\(g(f(x))-k(x)\text{.}\)
Write an expression for
\(f(g(k(x)))\text{.}\)
For what values of
\(x\) does
\(g(x)=k(x)\text{?}\)
For what values of
\(x\) is
\(f(x) \leq k(x)\text{?}\)
Refer to the function
\(g(x)=-\frac{1}{3}x+4\text{.}\)
Find a function equation for the inverse function,
\(g^{-1}\text{.}\)
What are the domain and range of
\(g^{-1}\text{?}\)
Graph
\(g(x)\) and
\(g^{-1}(x)\) on the same set of axes.
