Exponents, Logarithms, and Their Inverse Relationship - Algebra 2
Card 1 of 30
What is the inverse function of $f(x)=b^x$ (with $b>0$, $b\ne 1$) written as a log?
What is the inverse function of $f(x)=b^x$ (with $b>0$, $b\ne 1$) written as a log?
Tap to reveal answer
$f^{-1}(x)=\log_b(x)$. Exponential and logarithmic functions are inverses of each other.
$f^{-1}(x)=\log_b(x)$. Exponential and logarithmic functions are inverses of each other.
← Didn't Know|Knew It →
What is the inverse function of $g(x)=\log_b(x)$ (with valid base $b$) written exponentially?
What is the inverse function of $g(x)=\log_b(x)$ (with valid base $b$) written exponentially?
Tap to reveal answer
$g^{-1}(x)=b^x$. Logarithmic and exponential functions are inverses of each other.
$g^{-1}(x)=b^x$. Logarithmic and exponential functions are inverses of each other.
← Didn't Know|Knew It →
What is the exact solution of $\log_3(x)=\frac{1}{2}$?
What is the exact solution of $\log_3(x)=\frac{1}{2}$?
Tap to reveal answer
$x=\sqrt{3}$. Since $3^{1/2} = \sqrt{3}$.
$x=\sqrt{3}$. Since $3^{1/2} = \sqrt{3}$.
← Didn't Know|Knew It →
What is the exact value of $\log_8(4)$?
What is the exact value of $\log_8(4)$?
Tap to reveal answer
$\frac{2}{3}$. Since $8^{2/3} = (2^3)^{2/3} = 2^2 = 4$.
$\frac{2}{3}$. Since $8^{2/3} = (2^3)^{2/3} = 2^2 = 4$.
← Didn't Know|Knew It →
What is the exact value of $\log_{\frac{1}{4}}(2)$?
What is the exact value of $\log_{\frac{1}{4}}(2)$?
Tap to reveal answer
$-\frac{1}{2}$. Since $(\frac{1}{4})^{-1/2} = 4^{1/2} = 2$.
$-\frac{1}{2}$. Since $(\frac{1}{4})^{-1/2} = 4^{1/2} = 2$.
← Didn't Know|Knew It →
What is the exact value of $\log_{16}(\frac{1}{2})$?
What is the exact value of $\log_{16}(\frac{1}{2})$?
Tap to reveal answer
$-\frac{1}{4}$. Since $16^{-1/4} = (2^4)^{-1/4} = 2^{-1} = \frac{1}{2}$.
$-\frac{1}{4}$. Since $16^{-1/4} = (2^4)^{-1/4} = 2^{-1} = \frac{1}{2}$.
← Didn't Know|Knew It →
What is the exact solution of $\ln(x)=\ln(12)$?
What is the exact solution of $\ln(x)=\ln(12)$?
Tap to reveal answer
$x=12$. Equal natural logarithms means equal arguments.
$x=12$. Equal natural logarithms means equal arguments.
← Didn't Know|Knew It →
What is the exact solution of $\log_2(x-1)=3$?
What is the exact solution of $\log_2(x-1)=3$?
Tap to reveal answer
$x=9$. Since $2^3 = 8$, so $x - 1 = 8$.
$x=9$. Since $2^3 = 8$, so $x - 1 = 8$.
← Didn't Know|Knew It →
What is the exact solution of $\log_3(x^2)=2$ with $x>0$?
What is the exact solution of $\log_3(x^2)=2$ with $x>0$?
Tap to reveal answer
$x=3$. Using power rule: $2\log_3(x) = 2$, so $\log_3(x) = 1$.
$x=3$. Using power rule: $2\log_3(x) = 2$, so $\log_3(x) = 1$.
← Didn't Know|Knew It →
What is the exact solution of $4^x=2$?
What is the exact solution of $4^x=2$?
Tap to reveal answer
$x=\frac{1}{2}$. Since $4^{1/2} = 2$.
$x=\frac{1}{2}$. Since $4^{1/2} = 2$.
← Didn't Know|Knew It →
What is $\log_7(7)$?
What is $\log_7(7)$?
Tap to reveal answer
$1$. Any base raised to power 1 equals itself.
$1$. Any base raised to power 1 equals itself.
← Didn't Know|Knew It →
What is the exact solution of $\ln(x)=2$?
What is the exact solution of $\ln(x)=2$?
Tap to reveal answer
$x=e^2$. Converting logarithmic to exponential form: $e^2 = x$.
$x=e^2$. Converting logarithmic to exponential form: $e^2 = x$.
← Didn't Know|Knew It →
What is $\log_2(8)$?
What is $\log_2(8)$?
Tap to reveal answer
$3$. Since $2^3 = 8$.
$3$. Since $2^3 = 8$.
← Didn't Know|Knew It →
What is $\log_3(\frac{1}{9})$?
What is $\log_3(\frac{1}{9})$?
Tap to reveal answer
$-2$. Since $3^{-2} = \frac{1}{9}$.
$-2$. Since $3^{-2} = \frac{1}{9}$.
← Didn't Know|Knew It →
What is $\log_5(1)$?
What is $\log_5(1)$?
Tap to reveal answer
$0$. Any base raised to power 0 equals 1.
$0$. Any base raised to power 0 equals 1.
← Didn't Know|Knew It →
What is $\log_7(7)$?
What is $\log_7(7)$?
Tap to reveal answer
$1$. Any base raised to power 1 equals itself.
$1$. Any base raised to power 1 equals itself.
← Didn't Know|Knew It →
What is $\log_4(2)$?
What is $\log_4(2)$?
Tap to reveal answer
$\frac{1}{2}$. Since $4^{1/2} = \sqrt{4} = 2$.
$\frac{1}{2}$. Since $4^{1/2} = \sqrt{4} = 2$.
← Didn't Know|Knew It →
What is $\log_{\frac{1}{2}}(4)$?
What is $\log_{\frac{1}{2}}(4)$?
Tap to reveal answer
$-2$. Since $(\frac{1}{2})^{-2} = 2^2 = 4$.
$-2$. Since $(\frac{1}{2})^{-2} = 2^2 = 4$.
← Didn't Know|Knew It →
What is $\log_{10}(0.01)$?
What is $\log_{10}(0.01)$?
Tap to reveal answer
$-2$. Since $10^{-2} = 0.01$.
$-2$. Since $10^{-2} = 0.01$.
← Didn't Know|Knew It →
What is $\ln(1)$?
What is $\ln(1)$?
Tap to reveal answer
$0$. Any base raised to power 0 equals 1.
$0$. Any base raised to power 0 equals 1.
← Didn't Know|Knew It →
What is $\ln(e^3)$?
What is $\ln(e^3)$?
Tap to reveal answer
$3$. Natural logarithm and exponential are inverse functions.
$3$. Natural logarithm and exponential are inverse functions.
← Didn't Know|Knew It →
What is $e^{\ln(7)}$?
What is $e^{\ln(7)}$?
Tap to reveal answer
$7$. Exponential and natural logarithm are inverse functions.
$7$. Exponential and natural logarithm are inverse functions.
← Didn't Know|Knew It →
What is the exact solution of $2^x=16$?
What is the exact solution of $2^x=16$?
Tap to reveal answer
$x=4$. Since $2^4 = 16$.
$x=4$. Since $2^4 = 16$.
← Didn't Know|Knew It →
What is the exact solution of $3^x=\frac{1}{27}$?
What is the exact solution of $3^x=\frac{1}{27}$?
Tap to reveal answer
$x=-3$. Since $3^{-3} = \frac{1}{27}$.
$x=-3$. Since $3^{-3} = \frac{1}{27}$.
← Didn't Know|Knew It →
What is the exact solution of $10^x=0.001$?
What is the exact solution of $10^x=0.001$?
Tap to reveal answer
$x=-3$. Since $10^{-3} = 0.001$.
$x=-3$. Since $10^{-3} = 0.001$.
← Didn't Know|Knew It →
What is the exact solution of $\log_5(x)=3$?
What is the exact solution of $\log_5(x)=3$?
Tap to reveal answer
$x=125$. Since $5^3 = 125$.
$x=125$. Since $5^3 = 125$.
← Didn't Know|Knew It →
What is the exact solution of $\log_2(x)=-4$?
What is the exact solution of $\log_2(x)=-4$?
Tap to reveal answer
$x=\frac{1}{16}$. Since $2^{-4} = \frac{1}{16}$.
$x=\frac{1}{16}$. Since $2^{-4} = \frac{1}{16}$.
← Didn't Know|Knew It →
What is the exact solution of $\ln(x)=2$?
What is the exact solution of $\ln(x)=2$?
Tap to reveal answer
$x=e^2$. Converting logarithmic to exponential form: $e^2 = x$.
$x=e^2$. Converting logarithmic to exponential form: $e^2 = x$.
← Didn't Know|Knew It →
What is the exact solution of $e^x=\frac{1}{e^5}$?
What is the exact solution of $e^x=\frac{1}{e^5}$?
Tap to reveal answer
$x=-5$. Since $e^{-5} = \frac{1}{e^5}$.
$x=-5$. Since $e^{-5} = \frac{1}{e^5}$.
← Didn't Know|Knew It →
What is the exact solution of $4^x=2$?
What is the exact solution of $4^x=2$?
Tap to reveal answer
$x=\frac{1}{2}$. Since $4^{1/2} = 2$.
$x=\frac{1}{2}$. Since $4^{1/2} = 2$.
← Didn't Know|Knew It →