2.10 Inverses of exponential functions
Considering the general form of a logarithmic function f(x) = a \log_b x:
What do the constants a and b represent?
What conditions must be met for a and b?
Identify the values of a and b for each function:
f(x) = 3 \log_{5} x
g(x) = -2 \log x
h(x) = 0 \log_{3} x
j(x) = \log_{2} x
Is the function types has an inverse relationship between input and output values?
Quadratic functions
Exponential functions
Logarithmic functions
Linear functions
Identify if the statement is true or false.
The general form of a logarithmic function is f(x) = a \log_b x, where b>0, and a ≠0.
The functions f(x) = \log_bx and g(x) = b^x are inverse functions.
The logarithmic function f(x) = \log_b x, where b > 0 and b ≠1, reflects the exponential function g(x) = b^x over the line h(x) = x.
The output values of general-form exponential functions change proportionately as input values increase in equal-length intervals.
Given the functions f \left( x \right) = \log_b x and g \left( x \right) = b^x, what is the value of g(f(x)) and f(g(x))?
If a logarithmic function is reflected over the line y = x, what type of function will it become?
For each exponential function and corresponding solutions in the table, find the corresponding ordered pair that satisfies the related logarithmic function.
| Exponential function | Solution | Logarithmic function | Solution |
|---|---|---|---|
| g(t) = 2^t | (3, 8) | f(s) = \log_2 s | |
| g(t) = 3^t | (-1, \dfrac{1}{3}) | f(s) = \log_3 s | |
| g(t) = (0.5)^t | (2, 0.25) | f(s) = \log_{0.5} s |
Evaluate the functions at the specified values of x:
f(x) = 2 \log_{3} x at x=9
f(x) = \log_{2} x at x=8
f(x) = 2^x at x=4
f(x) = 3^{\frac{x}{4}} at x=28
Solve for x:
\log_5(x) = 3
\log_3 (x) = 2
2^x = 32
4^x = 64
Consider the exponential function y=5^x. How does the output value change as the input value increases by one unit?
Consider the logarithmic function y=log_5x. How does the input value change as the output value increases by one unit?
Find the inverse functions:
g(x) = 5^x
g(x) = \left(\dfrac{1}{2}\right)^x
f(x) = \log_{3} x
h(x) = \log_{5} (x - 4)
The graph represents the exponential function g(x) = 3^{x - 1}.
Sketch the graph of the corresponding logarithmic function f(x) = log_3 (x + 1), showing that it is a reflection of the exponential function over the line y = x.
If f(x) = \log_2 x and g(x) = 2^x, find the value of g(f(16)) and f(g(5)). Explain the relationship between the two functions.
Graph the logarithmic function f(x) = log_3 x and its reflection over the identity function on the same set of axes.
For each function:
Find the inverse of the function.
Given the points on the graphs of the respective functions, determine the corresponding points on their inverse functions.
f(x) = 2^x with point (2, 8)
g(x) = \log_2 x with point (4, 2)
Consider the exponential function y=10^x. If we know that the output value triples when the input value increases by one unit, can we determine the base of the exponential function? Explain your reasoning.
The half-life of a radioactive substance is 20 years.
Write an exponential function to model the decay of this substance over time.
Write the corresponding logarithmic function.
How many years it will take for the substance to decay to 25\% of its original amount?
A scientist is studying the growth of a bacteria culture. The graph shows the number of bacteria as a function of time, modeled by the function g(x)=2^x.
The scientist wants to predict the time it would take for the bacteria culture to reach a certain number.
Sketch the graph of the corresponding logarithmic function, f(x).
What is the estimated time it would take for the bacteria culture to reach a count of 1000? Explain your reasoning.