Logarithmic Functions - ACT Math
Card 1 of 30
Simplify: $\text{log}_b(1)$.
Simplify: $\text{log}_b(1)$.
Tap to reveal answer
$\text{log}_b(1) = 0$. Any base raised to the zero power equals 1.
$\text{log}_b(1) = 0$. Any base raised to the zero power equals 1.
← Didn't Know|Knew It →
Solve for $x$: $
log_2(x)=5$.
Solve for $x$: $ log_2(x)=5$.
Tap to reveal answer
$x=32$. Convert to exponential form: $x = 2^5 = 32$.
$x=32$. Convert to exponential form: $x = 2^5 = 32$.
← Didn't Know|Knew It →
Solve for $x$: $
log_5(x)=0$.
Solve for $x$: $ log_5(x)=0$.
Tap to reveal answer
$x=1$. Since $\log_5(1) = 0$, we have $x = 1$.
$x=1$. Since $\log_5(1) = 0$, we have $x = 1$.
← Didn't Know|Knew It →
What restriction on the base $b$ is required for $
log_b(x)$ to be a logarithm?
What restriction on the base $b$ is required for $ log_b(x)$ to be a logarithm?
Tap to reveal answer
$b>0$ and $b\neq 1$. Base must be positive and not equal to 1 for logarithm to be well-defined.
$b>0$ and $b\neq 1$. Base must be positive and not equal to 1 for logarithm to be well-defined.
← Didn't Know|Knew It →
Solve for $x$: $\text{log}_x(16) = 2$.
Solve for $x$: $\text{log}_x(16) = 2$.
Tap to reveal answer
$x = 4$. Convert to exponential form: $x^2 = 16$, so $x = 4$.
$x = 4$. Convert to exponential form: $x^2 = 16$, so $x = 4$.
← Didn't Know|Knew It →
Evaluate: $\text{log}_2(32)$.
Evaluate: $\text{log}_2(32)$.
Tap to reveal answer
$5$. Since $2^5 = 32$, the logarithm equals 5.
$5$. Since $2^5 = 32$, the logarithm equals 5.
← Didn't Know|Knew It →
What is the value of $\text{log}_e(e^3)$?
What is the value of $\text{log}_e(e^3)$?
Tap to reveal answer
$3$. Natural logarithm and exponential with same base cancel.
$3$. Natural logarithm and exponential with same base cancel.
← Didn't Know|Knew It →
Simplify: $\text{ln}(e^5)$.
Simplify: $\text{ln}(e^5)$.
Tap to reveal answer
$5$. Natural logarithm and exponential cancel each other.
$5$. Natural logarithm and exponential cancel each other.
← Didn't Know|Knew It →
Evaluate: $\text{log}_9(81)$.
Evaluate: $\text{log}_9(81)$.
Tap to reveal answer
$2$. Since $9^2 = 81$, the logarithm equals 2.
$2$. Since $9^2 = 81$, the logarithm equals 2.
← Didn't Know|Knew It →
Solve for $x$: $
log_5(x)=0$.
Solve for $x$: $ log_5(x)=0$.
Tap to reveal answer
$x=1$. Since $\log_5(1) = 0$, we have $x = 1$.
$x=1$. Since $\log_5(1) = 0$, we have $x = 1$.
← Didn't Know|Knew It →
What is the meaning of $
log(x)$ on the ACT when no base is written?
What is the meaning of $ log(x)$ on the ACT when no base is written?
Tap to reveal answer
$\log(x)=\log_{10}(x)$. When no base is specified, logarithm assumes base 10 (common logarithm).
$\log(x)=\log_{10}(x)$. When no base is specified, logarithm assumes base 10 (common logarithm).
← Didn't Know|Knew It →
Solve for $x$: $
log_2(x)+\log_2(x)=6$.
Solve for $x$: $ log_2(x)+\log_2(x)=6$.
Tap to reveal answer
$x=8$. Simplify to $\log_2(x^2) = 6$, so $x^2 = 2^6 = 64$, giving $x = 8$.
$x=8$. Simplify to $\log_2(x^2) = 6$, so $x^2 = 2^6 = 64$, giving $x = 8$.
← Didn't Know|Knew It →
What is $\ln(e^5)$?
What is $\ln(e^5)$?
Tap to reveal answer
$5$. Natural log and exponential with base $e$ are inverse functions.
$5$. Natural log and exponential with base $e$ are inverse functions.
← Didn't Know|Knew It →
What is the change-of-base formula for $\log_b(x)$ using base $10$?
What is the change-of-base formula for $\log_b(x)$ using base $10$?
Tap to reveal answer
$\log_b(x)=\frac{\log(x)}{\log(b)}$. Converts any base logarithm to common logarithm (base 10).
$\log_b(x)=\frac{\log(x)}{\log(b)}$. Converts any base logarithm to common logarithm (base 10).
← Didn't Know|Knew It →
What is $
log_{10}(1000)$?
What is $ log_{10}(1000)$?
Tap to reveal answer
$3$. Since $10^3 = 1000$, we have $\log_{10}(1000) = 3$.
$3$. Since $10^3 = 1000$, we have $\log_{10}(1000) = 3$.
← Didn't Know|Knew It →
If $\text{log}_5(x) = 0$, what is $x$?
If $\text{log}_5(x) = 0$, what is $x$?
Tap to reveal answer
$x = 1$. Any base raised to the zero power equals 1.
$x = 1$. Any base raised to the zero power equals 1.
← Didn't Know|Knew It →
Evaluate: $\text{ln}(1)$.
Evaluate: $\text{ln}(1)$.
Tap to reveal answer
$0$. Natural logarithm of 1 always equals zero.
$0$. Natural logarithm of 1 always equals zero.
← Didn't Know|Knew It →
Evaluate: $\text{ln}(e)$.
Evaluate: $\text{ln}(e)$.
Tap to reveal answer
$1$. Natural logarithm of its base equals 1.
$1$. Natural logarithm of its base equals 1.
← Didn't Know|Knew It →
Simplify: $\text{ln}(e^5)$.
Simplify: $\text{ln}(e^5)$.
Tap to reveal answer
$5$. Natural logarithm and exponential cancel each other.
$5$. Natural logarithm and exponential cancel each other.
← Didn't Know|Knew It →
If $\text{log}_b(2) = 0.301$, find $\text{log}_b(8)$.
If $\text{log}_b(2) = 0.301$, find $\text{log}_b(8)$.
Tap to reveal answer
$0.903$. Use power property: $\text{log}_b(8) = \text{log}_b(2^3) = 3 \times 0.301$.
$0.903$. Use power property: $\text{log}_b(8) = \text{log}_b(2^3) = 3 \times 0.301$.
← Didn't Know|Knew It →
What is the definition of a logarithm?
What is the definition of a logarithm?
Tap to reveal answer
If $b^x = y$, then $\text{log}_b(y) = x$. The logarithm is the inverse operation of exponentiation.
If $b^x = y$, then $\text{log}_b(y) = x$. The logarithm is the inverse operation of exponentiation.
← Didn't Know|Knew It →
State the logarithmic identity for $\text{log}_b(b^x)$.
State the logarithmic identity for $\text{log}_b(b^x)$.
Tap to reveal answer
$\text{log}_b(b^x) = x$. The base and its logarithm cancel each other out.
$\text{log}_b(b^x) = x$. The base and its logarithm cancel each other out.
← Didn't Know|Knew It →
What is the inverse function of $y = b^x$?
What is the inverse function of $y = b^x$?
Tap to reveal answer
$y = \text{log}_b(x)$. Logarithmic and exponential functions are inverse operations.
$y = \text{log}_b(x)$. Logarithmic and exponential functions are inverse operations.
← Didn't Know|Knew It →
Express $b^{\text{log}_b(x)}$ in simplest form.
Express $b^{\text{log}_b(x)}$ in simplest form.
Tap to reveal answer
$b^{\text{log}_b(x)} = x$. The exponential and logarithm with same base cancel out.
$b^{\text{log}_b(x)} = x$. The exponential and logarithm with same base cancel out.
← Didn't Know|Knew It →
What is the change of base formula for $\text{log}_b(x)$?
What is the change of base formula for $\text{log}_b(x)$?
Tap to reveal answer
$\text{log}_b(x) = \frac{\text{log}_k(x)}{\text{log}_k(b)}$. Allows conversion between different logarithmic bases.
$\text{log}_b(x) = \frac{\text{log}_k(x)}{\text{log}_k(b)}$. Allows conversion between different logarithmic bases.
← Didn't Know|Knew It →
Simplify: $\text{log}_b(b)$.
Simplify: $\text{log}_b(b)$.
Tap to reveal answer
$\text{log}_b(b) = 1$. A base raised to the first power equals itself.
$\text{log}_b(b) = 1$. A base raised to the first power equals itself.
← Didn't Know|Knew It →
What is the value of $\text{log}_a(a^x)$?
What is the value of $\text{log}_a(a^x)$?
Tap to reveal answer
$\text{log}_a(a^x) = x$. The logarithm and exponential with same base cancel.
$\text{log}_a(a^x) = x$. The logarithm and exponential with same base cancel.
← Didn't Know|Knew It →
Express $\text{log}_b(xy)$ using properties of logarithms.
Express $\text{log}_b(xy)$ using properties of logarithms.
Tap to reveal answer
$\text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y)$. The logarithm of a product equals the sum of logarithms.
$\text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y)$. The logarithm of a product equals the sum of logarithms.
← Didn't Know|Knew It →
Express $\text{log}_b(\frac{x}{y})$ using properties of logarithms.
Express $\text{log}_b(\frac{x}{y})$ using properties of logarithms.
Tap to reveal answer
$\text{log}_b(\frac{x}{y}) = \text{log}_b(x) - \text{log}_b(y)$. The logarithm of a quotient equals the difference of logarithms.
$\text{log}_b(\frac{x}{y}) = \text{log}_b(x) - \text{log}_b(y)$. The logarithm of a quotient equals the difference of logarithms.
← Didn't Know|Knew It →
Express $\text{log}_b(x^n)$ using properties of logarithms.
Express $\text{log}_b(x^n)$ using properties of logarithms.
Tap to reveal answer
$\text{log}_b(x^n) = n \times \text{log}_b(x)$. The exponent can be brought down as a coefficient.
$\text{log}_b(x^n) = n \times \text{log}_b(x)$. The exponent can be brought down as a coefficient.
← Didn't Know|Knew It →