Exponential Function Manipulation - AP Precalculus
Card 1 of 30
Express $b^{x+y}$ in its expanded form.
Express $b^{x+y}$ in its expanded form.
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$b^x \times b^y$. The sum rule for exponents breaks into separate factors.
$b^x \times b^y$. The sum rule for exponents breaks into separate factors.
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What is the value of $x$ if $2^x = 1$?
What is the value of $x$ if $2^x = 1$?
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$x = 0$. Any positive number raised to power 0 equals 1.
$x = 0$. Any positive number raised to power 0 equals 1.
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How do you solve $3^x = 27$?
How do you solve $3^x = 27$?
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$x = 3$. Since $3^3 = 27$, we have $x = 3$.
$x = 3$. Since $3^3 = 27$, we have $x = 3$.
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What is the domain of $f(x) = 2^x$?
What is the domain of $f(x) = 2^x$?
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$(-\text{inf}, \text{inf})$. Exponential functions accept all real number inputs.
$(-\text{inf}, \text{inf})$. Exponential functions accept all real number inputs.
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How do you solve $e^x = 5$?
How do you solve $e^x = 5$?
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$x = \text{ln}(5)$. Taking the natural logarithm of both sides isolates $x$.
$x = \text{ln}(5)$. Taking the natural logarithm of both sides isolates $x$.
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Evaluate $2^x = 32$ for $x$.
Evaluate $2^x = 32$ for $x$.
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$x = 5$. Since $2^5 = 32$, we have $x = 5$.
$x = 5$. Since $2^5 = 32$, we have $x = 5$.
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Identify the base in the function $f(x) = 3 \times 2^x$.
Identify the base in the function $f(x) = 3 \times 2^x$.
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- The base is the number being raised to the power $x$.
- The base is the number being raised to the power $x$.
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How do you express $\text{log}_a(a^x)$?
How do you express $\text{log}_a(a^x)$?
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$x$. The logarithm and exponential with the same base cancel out.
$x$. The logarithm and exponential with the same base cancel out.
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What is the simplified form of $\text{log}_b(b)$?
What is the simplified form of $\text{log}_b(b)$?
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$1$. The logarithm of a base to itself equals 1.
$1$. The logarithm of a base to itself equals 1.
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What is $b^1$ equal to?
What is $b^1$ equal to?
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$b$. Any number raised to the power 1 equals itself.
$b$. Any number raised to the power 1 equals itself.
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What is the expression for $e^{2\text{ln}(x)}$?
What is the expression for $e^{2\text{ln}(x)}$?
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$x^2$. Using the property $e^{\ln(x)} = x$ and power rule.
$x^2$. Using the property $e^{\ln(x)} = x$ and power rule.
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What is $\text{ln}(\text{e}^x)$?
What is $\text{ln}(\text{e}^x)$?
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$x$. Natural logarithm and $e$ are inverse functions.
$x$. Natural logarithm and $e$ are inverse functions.
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What is the inverse of $f(x) = \text{e}^x$?
What is the inverse of $f(x) = \text{e}^x$?
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$f^{-1}(x) = \text{ln}(x)$. The natural exponential and logarithm are inverse functions.
$f^{-1}(x) = \text{ln}(x)$. The natural exponential and logarithm are inverse functions.
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What is the base $b$ in $f(x) = b^x$ for $b > 0$ and $b \neq 1$?
What is the base $b$ in $f(x) = b^x$ for $b > 0$ and $b \neq 1$?
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$b$ is a positive constant not equal to 1. Base must be positive and not equal to 1 for exponential functions.
$b$ is a positive constant not equal to 1. Base must be positive and not equal to 1 for exponential functions.
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Evaluate $3^x = 243$ for $x$.
Evaluate $3^x = 243$ for $x$.
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$x = 5$. Since $3^5 = 243$, we have $x = 5$.
$x = 5$. Since $3^5 = 243$, we have $x = 5$.
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Rewrite the expression $b^{-x}$ in terms of fractions.
Rewrite the expression $b^{-x}$ in terms of fractions.
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$\frac{1}{b^x}$. Negative exponents represent reciprocals of positive exponents.
$\frac{1}{b^x}$. Negative exponents represent reciprocals of positive exponents.
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What is $\text{log}_a(1)$ equal to?
What is $\text{log}_a(1)$ equal to?
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$0$. Any number raised to the power 0 equals 1, so $\log_a(1) = 0$.
$0$. Any number raised to the power 0 equals 1, so $\log_a(1) = 0$.
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What is $f(x) = 2^x$ expressed as a logarithmic function?
What is $f(x) = 2^x$ expressed as a logarithmic function?
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$x = \text{log}_2(f(x))$. Taking the logarithm base 2 of both sides isolates $x$.
$x = \text{log}_2(f(x))$. Taking the logarithm base 2 of both sides isolates $x$.
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What is the expression for $e^{2\text{ln}(x)}$?
What is the expression for $e^{2\text{ln}(x)}$?
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$x^2$. Using the property $e^{\ln(x)} = x$ and power rule.
$x^2$. Using the property $e^{\ln(x)} = x$ and power rule.
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What does the expression $b^{x-y}$ simplify to?
What does the expression $b^{x-y}$ simplify to?
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$\frac{b^x}{b^y}$. Subtracting exponents equals division of the same base powers.
$\frac{b^x}{b^y}$. Subtracting exponents equals division of the same base powers.
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Which rule is used to solve $b^{x+y} = b^x \times b^y$?
Which rule is used to solve $b^{x+y} = b^x \times b^y$?
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Exponential rule for addition. This is the fundamental property for combining exponential expressions.
Exponential rule for addition. This is the fundamental property for combining exponential expressions.
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What is the derivative of $f(x) = e^x$?
What is the derivative of $f(x) = e^x$?
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$f'(x) = e^x$. The natural exponential function is its own derivative.
$f'(x) = e^x$. The natural exponential function is its own derivative.
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What is $\text{log}_b(b^x)$ equal to?
What is $\text{log}_b(b^x)$ equal to?
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$x$. The logarithm and exponential with the same base cancel out.
$x$. The logarithm and exponential with the same base cancel out.
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How do you rewrite $4^x = 16$ using logarithms?
How do you rewrite $4^x = 16$ using logarithms?
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$x = \text{log}_4(16)$. Converting exponential to logarithmic form by taking log of both sides.
$x = \text{log}_4(16)$. Converting exponential to logarithmic form by taking log of both sides.
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What is the inverse of the exponential function $f(x) = a^x$?
What is the inverse of the exponential function $f(x) = a^x$?
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$f^{-1}(x) = \text{log}_a(x)$. Exponential and logarithmic functions are inverse pairs.
$f^{-1}(x) = \text{log}_a(x)$. Exponential and logarithmic functions are inverse pairs.
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What property of exponents is used in $a^x \times a^y = a^{x+y}$?
What property of exponents is used in $a^x \times a^y = a^{x+y}$?
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Product of powers. When multiplying powers with the same base, add the exponents.
Product of powers. When multiplying powers with the same base, add the exponents.
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How do you express $e^{\text{ln}(x)}$ in terms of $x$?
How do you express $e^{\text{ln}(x)}$ in terms of $x$?
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$x$. The exponential and natural logarithm functions are inverses.
$x$. The exponential and natural logarithm functions are inverses.
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What is the value of $f(x) = 5 \times 2^x$ when $x = 3$?
What is the value of $f(x) = 5 \times 2^x$ when $x = 3$?
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$40$. Substitute $x = 3$: $f(3) = 5 \times 2^3 = 5 \times 8 = 40$.
$40$. Substitute $x = 3$: $f(3) = 5 \times 2^3 = 5 \times 8 = 40$.
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Identify the base in the function $f(x) = 3 \times 2^x$.
Identify the base in the function $f(x) = 3 \times 2^x$.
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- The base is the number being raised to the power $x$.
- The base is the number being raised to the power $x$.
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What is the exponential form of $x = \text{log}_b(y)$?
What is the exponential form of $x = \text{log}_b(y)$?
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$b^x = y$. Converting from logarithmic to exponential form.
$b^x = y$. Converting from logarithmic to exponential form.
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