Precalculus Practice: Exponential, Logarithmic, and Function Operations

Q1. Find the inverse of the function $f(x) = 5 \cdot 10^x$.

Background

Topic: Inverse Functions and Logarithms

This question tests your understanding of how to find the inverse of an exponential function, which often involves using logarithms to solve for the original input variable.

Key Terms and Formulas

• Inverse function: If $f(x)$ is a function, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$.

• Logarithm: The logarithm base $b$ of $a$ is the exponent to which $b$ must be raised to get $a$.

• Exponential and logarithmic relationship: $y = b^x \iff x = \log_b(y)$

Step-by-Step Guidance

1. Start by replacing $f(x)$ with $y$ for clarity: $y = 5 \cdot 10^x$.

2. Switch $x$ and $y$ to begin finding the inverse: $x = 5 \cdot 10^y$.

3. Isolate $10^y$ by dividing both sides by $5$:$\frac{x}{5} = 10^y$.

4. Take the logarithm (base 10) of both sides to solve for $y$: $\log\left(\frac{x}{5}\right) = y$.

Try solving on your own before revealing the answer!

Q2. Given $f(x) = x + 1$ and $g(x) = \frac{1}{x-2}$, find $(f \circ g)(5)$.

Background

Topic: Function Composition

This question tests your ability to compose two functions and evaluate the result at a specific value.

Key Terms and Formulas

• Function composition: $(f \circ g)(x) = f(g(x))$

Step-by-Step Guidance

1. First, evaluate $g(5)$ by substituting $x = 5$ into $g(x)$: $g(5) = \frac{1}{5-2}$.

2. Next, use the result from $g(5)$ as the input for $f(x)$: $f(g(5)) = f\left(\frac{1}{3}\right)$.

3. Recall that $f(x) = x + 1$, so substitute $x = \frac{1}{3}$ into $f(x)$.

Try solving on your own before revealing the answer!

Q3. Solve the logarithmic equation for $x$: $\log(x+3) - \log(x) = 1$.

Background

Topic: Logarithmic Equations

This question tests your ability to use properties of logarithms to solve for a variable.

Key Terms and Formulas

• Logarithm subtraction property: $\log_a(M) - \log_a(N) = \log_a\left(\frac{M}{N}\right)$

• Exponentiation: If $\log_a(B) = C$, then $B = a^C$

Step-by-Step Guidance

1. Combine the logarithms using the subtraction property: $\log(x+3) - \log(x) = \log\left(\frac{x+3}{x}\right)$.

2. Set the combined logarithm equal to $1$:$\log\left(\frac{x+3}{x}\right) = 1$.

3. Rewrite the equation in exponential form: $\frac{x+3}{x} = 10^1$.

4. Solve for $x$ by multiplying both sides by $x$ and rearranging the equation.

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Q4. Suppose $y$ is increasing exponentially, with $y(0) = 5$ and $y(3) = 20$. Find $y(9)$.

Background

Topic: Exponential Growth Functions

This question tests your ability to model exponential growth and use given data points to find unknown values.

Key Terms and Formulas

• Exponential growth model: $y = c \cdot a^x$

• Initial value: $y(0) = c$

Step-by-Step Guidance

1. Use $y(0) = 5$ to determine $c$ in the model $y = c \cdot a^x$.

2. Use $y(3) = 20$ to set up the equation $20 = 5 \cdot a^3$ and solve for $a$.

3. Once you have $a$, write the full exponential model for $y(x)$.

4. Substitute $x = 9$ into your model to set up the calculation for $y(9)$.

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Q5. Solve for $x$: $\log(102x) = 12$

Background

Topic: Solving Logarithmic Equations

This question tests your ability to solve for a variable inside a logarithmic expression.

Key Terms and Formulas

• Logarithm definition: $\log_b(A) = C \iff A = b^C$

Step-by-Step Guidance

1. Recognize that $\log(102x)$ is shorthand for $\log_{10}(102x)$.

2. Rewrite the equation in exponential form: $102x = 10^{12}$.

3. Solve for $x$ by dividing both sides by $102$.

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Q6. Suppose $250.00 is deposited in an account with an interest rate of 2% compounded continuously. How long will it take for the money to quadruple?

Background

Topic: Exponential Growth and Continuous Compounding

This question tests your understanding of the continuous compounding formula and how to use logarithms to solve for time.

Key Terms and Formulas

• Continuous compounding formula: $A(t) = A_0 e^{kt}$

• Quadruple: Final amount is four times the initial amount.

• Natural logarithm: $\ln(x)$ is the logarithm base $e$.

Step-by-Step Guidance

1. Set up the equation for quadrupling: $4 \cdot 250 = 250 e^{0.02t}$.

2. Simplify both sides to get $4 = e^{0.02t}$.

3. Take the natural logarithm of both sides: $\ln(4) = 0.02t$.

4. Rearrange to solve for $t$ by dividing both sides by $0.02$.

Try solving on your own before revealing the answer!