Logarithmic Expressions - AP Precalculus
Card 1 of 30
What is the condensed form of $\log_2(x)+\log_2(y)-\log_2(4)$?
What is the condensed form of $\log_2(x)+\log_2(y)-\log_2(4)$?
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$\log_2\left(\frac{xy}{4}\right)$. Product property combines logs; quotient property for subtraction.
$\log_2\left(\frac{xy}{4}\right)$. Product property combines logs; quotient property for subtraction.
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What is the simplified value: $10^{\log(3)}$ equals what?
What is the simplified value: $10^{\log(3)}$ equals what?
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$3$. $10$ and $\log$ are inverses, so they cancel.
$3$. $10$ and $\log$ are inverses, so they cancel.
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Identify the simplified value: $\log_3\left(\frac{1}{27}\right)$ equals what?
Identify the simplified value: $\log_3\left(\frac{1}{27}\right)$ equals what?
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$-3$. $\frac{1}{27} = 3^{-3}$, so $\log_3(\frac{1}{27}) = -3$.
$-3$. $\frac{1}{27} = 3^{-3}$, so $\log_3(\frac{1}{27}) = -3$.
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What is the expanded form of $\log_5\left(\frac{25x^3}{y}\right)$?
What is the expanded form of $\log_5\left(\frac{25x^3}{y}\right)$?
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$2+3\log_5(x)-\log_5(y)$. Apply quotient, power, and $\log_5(25) = 2$.
$2+3\log_5(x)-\log_5(y)$. Apply quotient, power, and $\log_5(25) = 2$.
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What is the definition of a logarithm: $
log_b(a)$ equals what exponential statement?
What is the definition of a logarithm: $ log_b(a)$ equals what exponential statement?
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$\log_b(a)=c \iff b^c=a$. A logarithm asks: what power of $b$ gives $a$?
$\log_b(a)=c \iff b^c=a$. A logarithm asks: what power of $b$ gives $a$?
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What is the domain condition for $
log_b(a)$ in real numbers (for $a$ and $b$)?
What is the domain condition for $ log_b(a)$ in real numbers (for $a$ and $b$)?
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$a>0,\ b>0,\ b\ne 1$. Base and argument must be positive; base cannot equal 1.
$a>0,\ b>0,\ b\ne 1$. Base and argument must be positive; base cannot equal 1.
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What is the product property of logarithms for $\log_b(MN)$?
What is the product property of logarithms for $\log_b(MN)$?
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$\log_b(MN)=\log_b(M)+\log_b(N)$. The log of a product equals the sum of the logs.
$\log_b(MN)=\log_b(M)+\log_b(N)$. The log of a product equals the sum of the logs.
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Find the exact value of $\log_4(\sqrt{2})$.
Find the exact value of $\log_4(\sqrt{2})$.
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$\frac{1}{4}$. $\sqrt{2} = 2^{1/2} = 4^{1/4}$, so answer is $\frac{1}{4}$.
$\frac{1}{4}$. $\sqrt{2} = 2^{1/2} = 4^{1/4}$, so answer is $\frac{1}{4}$.
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What is the simplified value: $\ln(e^{-4})$ equals what?
What is the simplified value: $\ln(e^{-4})$ equals what?
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$-4$. $\ln$ and $e$ are inverse functions.
$-4$. $\ln$ and $e$ are inverse functions.
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What is the simplified value: $\log_7(7^{x-2})$ equals what?
What is the simplified value: $\log_7(7^{x-2})$ equals what?
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$x-2$. Log and exponential with same base cancel.
$x-2$. Log and exponential with same base cancel.
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What is the quotient property of logarithms for $\log_b\left(\frac{M}{N}\right)$?
What is the quotient property of logarithms for $\log_b\left(\frac{M}{N}\right)$?
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$\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)$. The log of a quotient equals the difference of the logs.
$\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)$. The log of a quotient equals the difference of the logs.
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What is the power property of logarithms for $\log_b(M^p)$?
What is the power property of logarithms for $\log_b(M^p)$?
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$\log_b(M^p)=p\log_b(M)$. Exponents come out front as coefficients.
$\log_b(M^p)=p\log_b(M)$. Exponents come out front as coefficients.
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What is the change-of-base formula for $\log_b(a)$ using base $k$?
What is the change-of-base formula for $\log_b(a)$ using base $k$?
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$\log_b(a)=\frac{\log_k(a)}{\log_k(b)}$. Convert between bases by dividing logs of the same base.
$\log_b(a)=\frac{\log_k(a)}{\log_k(b)}$. Convert between bases by dividing logs of the same base.
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Identify the simplified value: $\log_2(32)$ equals what?
Identify the simplified value: $\log_2(32)$ equals what?
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$5$. $32 = 2^5$, so $\log_2(32) = 5$.
$5$. $32 = 2^5$, so $\log_2(32) = 5$.
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What is the natural logarithm notation: $\ln(x)$ means $\log_b(x)$ with what base?
What is the natural logarithm notation: $\ln(x)$ means $\log_b(x)$ with what base?
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$\ln(x)=\log_e(x)$. Natural log has base $e$ (Euler's number).
$\ln(x)=\log_e(x)$. Natural log has base $e$ (Euler's number).
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What is the common logarithm notation: $\log(x)$ means $\log_b(x)$ with what base?
What is the common logarithm notation: $\log(x)$ means $\log_b(x)$ with what base?
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$\log(x)=\log_{10}(x)$. Common log has base 10 by convention.
$\log(x)=\log_{10}(x)$. Common log has base 10 by convention.
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What is $\log_b(1)$ for any valid base $b$?
What is $\log_b(1)$ for any valid base $b$?
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$\log_b(1)=0$. Any base raised to power 0 equals 1.
$\log_b(1)=0$. Any base raised to power 0 equals 1.
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What is $\log_b(b)$ for any valid base $b$?
What is $\log_b(b)$ for any valid base $b$?
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$\log_b(b)=1$. Any base raised to power 1 equals itself.
$\log_b(b)=1$. Any base raised to power 1 equals itself.
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What is the inverse property: $\log_b(b^x)$ equals what (for real $x$)?
What is the inverse property: $\log_b(b^x)$ equals what (for real $x$)?
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$\log_b(b^x)=x$. Log and exponential functions are inverses.
$\log_b(b^x)=x$. Log and exponential functions are inverses.
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What is the inverse property: $b^{\log_b(x)}$ equals what (for $x>0$)?
What is the inverse property: $b^{\log_b(x)}$ equals what (for $x>0$)?
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$b^{\log_b(x)}=x$. Exponential and log functions cancel each other.
$b^{\log_b(x)}=x$. Exponential and log functions cancel each other.
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Solve for $x$: $\log_4(x-1)=2$.
Solve for $x$: $\log_4(x-1)=2$.
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$x=17$. Convert to exponential: $x-1=4^2=16$, so $x=17$.
$x=17$. Convert to exponential: $x-1=4^2=16$, so $x=17$.
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What is the domain of $f(x)=\log_5(2x-3)$?
What is the domain of $f(x)=\log_5(2x-3)$?
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$x>\frac{3}{2}$. Set $2x-3>0$ and solve for $x$.
$x>\frac{3}{2}$. Set $2x-3>0$ and solve for $x$.
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What is the domain of $g(x)=\log_3(x^2-9)$?
What is the domain of $g(x)=\log_3(x^2-9)$?
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$x<-3$ or $x>3$. Set $x^2-9>0$; factor as $(x-3)(x+3)>0$.
$x<-3$ or $x>3$. Set $x^2-9>0$; factor as $(x-3)(x+3)>0$.
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What is the value of $\log_b(1)$ for any valid base $b$?
What is the value of $\log_b(1)$ for any valid base $b$?
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$\log_b(1)=0$. Because $b^0=1$ for any valid base $b$.
$\log_b(1)=0$. Because $b^0=1$ for any valid base $b$.
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What is the value of $\log_b(b)$ for any valid base $b$?
What is the value of $\log_b(b)$ for any valid base $b$?
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$\log_b(b)=1$. Because $b^1=b$ for any valid base $b$.
$\log_b(b)=1$. Because $b^1=b$ for any valid base $b$.
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What is the one-to-one property of logarithms?
What is the one-to-one property of logarithms?
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$\log_b(M)=\log_b(N)\iff M=N$. Equal logs with same base imply equal arguments.
$\log_b(M)=\log_b(N)\iff M=N$. Equal logs with same base imply equal arguments.
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Identify the expanded form of $\log_b!\left(\frac{x^3y}{\sqrt{z}}\right)$.
Identify the expanded form of $\log_b!\left(\frac{x^3y}{\sqrt{z}}\right)$.
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$3\log_b(x)+\log_b(y)-\frac{1}{2}\log_b(z)$. Apply product, quotient, and power rules; $\sqrt{z}=z^{1/2}$.
$3\log_b(x)+\log_b(y)-\frac{1}{2}\log_b(z)$. Apply product, quotient, and power rules; $\sqrt{z}=z^{1/2}$.
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Identify the condensed form of $2\log_b(x)-\log_b(y)+3\log_b(z)$.
Identify the condensed form of $2\log_b(x)-\log_b(y)+3\log_b(z)$.
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$\log_b!\left(\frac{x^2z^3}{y}\right)$. Combine using power rule, then product/quotient rules.
$\log_b!\left(\frac{x^2z^3}{y}\right)$. Combine using power rule, then product/quotient rules.
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What is the exact value of $\log_2(32)$?
What is the exact value of $\log_2(32)$?
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$5$. Since $2^5=32$, the answer is 5.
$5$. Since $2^5=32$, the answer is 5.
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What is the exact value of $\log_5!\left(\frac{1}{25}\right)$?
What is the exact value of $\log_5!\left(\frac{1}{25}\right)$?
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$-2$. Since $5^{-2}=\frac{1}{25}$, the answer is -2.
$-2$. Since $5^{-2}=\frac{1}{25}$, the answer is -2.
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