ACT Math Flashcards: Logarithmic Functions

What this deck covers

This deck focuses on Logarithmic Functions, giving you a quick way to review the definitions, rules, and examples that matter most for ACT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Express $b^{\text{log}_b(x)}$ in simplest form.

Answer: $b^{\text{log}_b(x)} = x$. The exponential and logarithm with same base cancel out.

Flashcard 2: What is $log_{10}(1000)$?

Answer: $3$. Since $10^3 = 1000$, we have $\log_{10}(1000) = 3$.

Flashcard 3: What is the product rule for logarithms: $log_b(MN)$ equals what?

Answer: $\log_b(MN)=\log_b(M)+\log_b(N)$. Product inside log equals sum of individual logs.

Flashcard 4: Express $\text{log}_b(xy)$ using properties of logarithms.

Answer: $\text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y)$. The logarithm of a product equals the sum of logarithms.

Flashcard 5: What is the identity value of $log_b(b)$ for any valid base $b$?

Answer: $\log_b(b)=1$. Base raised to power 1 equals itself, so $\log_b(b) = 1$.

Flashcard 6: What is $\ln(e^5)$?

Answer: $5$. Natural log and exponential with base $e$ are inverse functions.

Flashcard 7: What vertical asymptote does $f(x)=\log_b(x-h)+k$ have?

Answer: Vertical asymptote: $x=h$. Horizontal shift moves the vertical asymptote to $x = h$.

Flashcard 8: Solve for $x$: $log_2(x)=5$.

Answer: $x=32$. Convert to exponential form: $x = 2^5 = 32$.

Flashcard 9: What restriction on the input $x$ is required for $log_b(x)$ to be defined over the reals?

Answer: $x>0$. Input must be positive since logarithms of non-positive numbers are undefined in reals.

Flashcard 10: What is the value of $\text{log}_e(e^3)$?

Answer: $3$. Natural logarithm and exponential with same base cancel.

Flashcard 11: What vertical asymptote does $f(x)=\log_b(x)$ have?

Answer: Vertical asymptote: $x=0$. Function approaches negative infinity as $x$ approaches 0 from the right.

Flashcard 12: Convert $\text{ln}(x) = 2$ to exponential form.

Answer: $x = e^2$. Converting from logarithmic to exponential form.

Flashcard 13: If $\text{log}_b(10) = 1.5$, what is $\text{log}_b(100)$?

Answer: $3$. Use power property: $\text{log}_b(100) = \text{log}_b(10^2) = 2 \times 1.5$.

Flashcard 14: What is the quotient rule for logarithms: $\log_b\left(\frac{M}{N}\right)$ equals what?

Answer: $\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)$. Quotient inside log equals difference of individual logs.

Flashcard 15: Evaluate: $\text{log}_2(32)$.

Answer: $5$. Since $2^5 = 32$, the logarithm equals 5.

Flashcard 16: Solve for $x$: $log_2(x)+\log_2(4)=6$.

Answer: $x=16$. Use product rule: $\log_2(4x) = 6$, so $4x = 2^6 = 64$, giving $x = 16$.

Flashcard 17: What is the identity value of $log_b(b)$ for any valid base $b$?

Answer: $\log_b(b)=1$. Base raised to power 1 equals itself, so $\log_b(b) = 1$.

Flashcard 18: What is the range of the logarithmic function $f(x)=\log_b(x)$?

Answer: Range: all real numbers. Logarithm can output any real value as input varies over positive reals.

Flashcard 19: What is the value of $\text{log}_a(a^x)$?

Answer: $\text{log}_a(a^x) = x$. The logarithm and exponential with same base cancel.

Flashcard 20: What is the identity value of $log_b(1)$ for any valid base $b$?

Answer: $\log_b(1)=0$. Any base raised to power 0 equals 1, so $\log_b(1) = 0$.

Flashcard 21: What is $log_2(8)$?

Answer: $3$. Since $2^3 = 8$, we have $\log_2(8) = 3$.

Flashcard 22: What is the meaning of $\ln(x)$?

Answer: $\ln(x)=\log_e(x)$. Natural logarithm uses base $e$ (Euler's number).

Flashcard 23: What is the domain of the logarithmic function $f(x)=\log_b(x)$?

Answer: Domain: $x>0$. Input must be positive for logarithm to be defined in real numbers.

Flashcard 24: What is the inverse relationship between $b^x$ and $log_b(x)$?

Answer: $b^{\log_b(x)}=x$ and $\log_b(b^x)=x$. Exponential and logarithm functions are inverse operations that cancel each other.

Flashcard 25: Evaluate: $\text{log}_5(125)$.

Answer: $3$. Since $5^3 = 125$, the logarithm equals 3.

Flashcard 26: What is the change-of-base formula for $\log_b(x)$ using base $10$?

Answer: $\log_b(x)=\frac{\log(x)}{\log(b)}$. Converts any base logarithm to common logarithm (base 10).

Flashcard 27: What is the domain of $f(x)=\log_b(ax+c)$ in one inequality (assume $a\neq 0$)?

Answer: Domain: $ax+c>0$. Argument must be positive, so $ax + c > 0$ determines the domain.

Flashcard 28: What is the power rule for logarithms: $log_b(M^k)$ equals what?

Answer: $\log_b(M^k)=k\log_b(M)$. Power inside log becomes coefficient multiplying the log.

Flashcard 29: What is the base of natural logarithms?

Answer: The base is $e$, where $e \thickapprox 2.718$. Natural logarithms use Euler's number as the base.

Flashcard 30: Solve for $x$: $10^{\text{log}(x)} = 100$.

Answer: $x = 100$. Since $10^{\text{log}(x)} = x$, we have $x = 100$.

Flashcard 31: Find the domain of $f(x)=\log_7(2x+3)$ as an inequality in $x$.

Answer: Domain: $x>\frac{-3}{2}$. Argument $2x + 3$ must be positive: $2x + 3 > 0$ gives $x > -\frac{3}{2}$.

Flashcard 32: What is $log_3\left(\frac{1}{9}\right)$?

Answer: $-2$. Since $3^{-2} = \frac{1}{9}$, we have $\log_3\left(\frac{1}{9}\right) = -2$.

Flashcard 33: What vertical asymptote does $f(x)=\log_b(x)$ have?

Answer: Vertical asymptote: $x=0$. Function approaches negative infinity as $x$ approaches 0 from the right.

Flashcard 34: If $\text{log}_5(x) = 0$, what is $x$?

Answer: $x = 1$. Any base raised to the zero power equals 1.

Flashcard 35: Evaluate: $\text{log}_4(64)$.

Answer: $3$. Since $4^3 = 64$, the logarithm equals 3.

Flashcard 36: Expand $log_3\left(\frac{x^2}{y}\right)$ using log rules.

Answer: $\log_3\left(\frac{x^2}{y}\right)=2\log_3(x)-\log_3(y)$. Apply quotient rule, then power rule to expand completely.

Flashcard 37: Evaluate: $\text{ln}(1)$.

Answer: $0$. Natural logarithm of 1 always equals zero.

Flashcard 38: What is the $x$-intercept of $f(x)=\log_b(x)$ for any valid base $b$?

Answer: $x$-intercept: $(1,0)$. Graph crosses x-axis where $\log_b(x) = 0$, which occurs at $x = 1$.

Flashcard 39: If $a = \text{log}_b(x)$, express $x$ in terms of $b$ and $a$.

Answer: $x = b^a$. Converting from logarithmic to exponential form.

Flashcard 40: Find the domain of $f(x)=\log_2(x-5)$ as an inequality in $x$.

Answer: Domain: $x>5$. Argument $x - 5$ must be positive for logarithm to be defined.

Flashcard 41: Convert $\text{ln}(x) = 2$ to exponential form.

Answer: $x = e^2$. Converting from logarithmic to exponential form.

Flashcard 42: Solve for $x$: $\text{log}_x(16) = 2$.

Answer: $x = 4$. Convert to exponential form: $x^2 = 16$, so $x = 4$.

Flashcard 43: Identify the property: $n \times \text{log}_b(x) = \text{log}_b(x^n)$.

Answer: Power Property. Used when raising logarithmic arguments to powers.

Flashcard 44: Evaluate: $\text{ln}(1)$.

Answer: $0$. Natural logarithm of 1 always equals zero.

Flashcard 45: Evaluate: $\text{log}_2(1)$.

Answer: $0$. Any logarithm of 1 equals zero.

Flashcard 46: Evaluate: $\text{log}_4(64)$.

Answer: $3$. Since $4^3 = 64$, the logarithm equals 3.

Flashcard 47: Convert $\text{log}_b(x) = y$ to exponential form.

Answer: $b^y = x$. Converting from logarithmic to exponential form.

Flashcard 48: What is the change of base formula for $\text{log}_b(x)$?

Answer: $\text{log}_b(x) = \frac{\text{log}_k(x)}{\text{log}_k(b)}$. Allows conversion between different logarithmic bases.

Flashcard 49: If $\text{log}_b(x) = 2$ and $\text{log}_b(y) = 3$, find $\text{log}_b(xy)$.

Answer: $5$. Use product property: $\text{log}_b(xy) = 2 + 3 = 5$.

Flashcard 50: Find the domain of $f(x)=\log_7(2x+3)$ as an inequality in $x$.

Answer: Domain: $x>\frac{-3}{2}$. Argument $2x + 3$ must be positive: $2x + 3 > 0$ gives $x > -\frac{3}{2}$.

Flashcard 51: Solve for $x$: $log_3(x)=\log_3(27)$.

Answer: $x=27$. If logarithms are equal, their arguments must be equal.

Flashcard 52: What transformation gives the domain condition for $f(x)=\log_b(x-h)+k$?

Answer: Domain: $x>h$. Horizontal shift by $h$ units moves the domain restriction accordingly.

Flashcard 53: Express $\text{log}_b(\frac{x}{y})$ using properties of logarithms.

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.

Flashcard 54: Solve for $x$: $log_4(x-1)=2$.

Answer: $x=17$. Convert to exponential: $x - 1 = 4^2 = 16$, so $x = 17$.

Flashcard 55: What is the definition of a logarithm?

Answer: If $b^x = y$, then $\text{log}_b(y) = x$. The logarithm is the inverse operation of exponentiation.

Flashcard 56: What is the base of natural logarithms?

Answer: The base is $e$, where $e \thickapprox 2.718$. Natural logarithms use Euler's number as the base.

Flashcard 57: Solve for $x$: $\text{log}_x(16) = 2$.

Answer: $x = 4$. Convert to exponential form: $x^2 = 16$, so $x = 4$.

Flashcard 58: Identify the property: $\text{log}_b(x) - \text{log}_b(y) = \text{log}_b(\frac{x}{y})$.

Answer: Quotient Property. Used when dividing arguments inside logarithms.

Flashcard 59: Find the domain of $f(x)=\log_2(x-5)$ as an inequality in $x$.

Answer: Domain: $x>5$. Argument $x - 5$ must be positive for logarithm to be defined.

Flashcard 60: Simplify: $\text{ln}(e^5)$.

Answer: $5$. Natural logarithm and exponential cancel each other.

Flashcard 61: Express $\text{log}_b(xy)$ using properties of logarithms.

Answer: $\text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y)$. The logarithm of a product equals the sum of logarithms.

Flashcard 62: Solve for $x$: $log_3(x)-\log_3(9)=1$.

Answer: $x=27$. Use quotient rule: $\log_3\left(\frac{x}{9}\right) = 1$, so $\frac{x}{9} = 3^1 = 3$.

Flashcard 63: What is the change-of-base formula for $\log_b(x)$ using base $e$?

Answer: $\log_b(x)=\frac{\ln(x)}{\ln(b)}$. Converts any base logarithm to natural logarithm (base $e$.)

Flashcard 64: Simplify: $\text{ln}(e^5)$.

Answer: $5$. Natural logarithm and exponential cancel each other.

Flashcard 65: Express $\text{log}_b(x^n)$ using properties of logarithms.

Answer: $\text{log}_b(x^n) = n \times \text{log}_b(x)$. The exponent can be brought down as a coefficient.

Flashcard 66: What is the power rule for logarithms: $log_b(M^k)$ equals what?

Answer: $\log_b(M^k)=k\log_b(M)$. Power inside log becomes coefficient multiplying the log.

Flashcard 67: Solve for $x$: $log_5(x)=0$.

Answer: $x=1$. Since $\log_5(1) = 0$, we have $x = 1$.

Flashcard 68: Convert $\text{log}_b(x) = y$ to exponential form.

Answer: $b^y = x$. Converting from logarithmic to exponential form.

Flashcard 69: What is the range of the logarithmic function $f(x)=\log_b(x)$?

Answer: Range: all real numbers. Logarithm can output any real value as input varies over positive reals.

Flashcard 70: Simplify using properties: $\text{log}_b(b^{-3})$.

Answer: $-3$. The logarithm and exponential with same base cancel.

Flashcard 71: Solve for $x$: $log_2(x)+\log_2(4)=6$.

Answer: $x=16$. Use product rule: $\log_2(4x) = 6$, so $4x = 2^6 = 64$, giving $x = 16$.

Flashcard 72: What is the change-of-base formula for $log_b(x)$ using base $e$?

Answer: $\log_b(x)=\frac{\ln(x)}{\ln(b)}$. Converts any base logarithm to natural logarithm (base $e$).

Flashcard 73: Simplify: $\text{log}_b(b)$.

Answer: $\text{log}_b(b) = 1$. A base raised to the first power equals itself.

Flashcard 74: Express $\text{log}_b(x^n)$ using properties of logarithms.

Answer: $\text{log}_b(x^n) = n \times \text{log}_b(x)$. The exponent can be brought down as a coefficient.

Flashcard 75: Condense $2\log_b(x)-3\log_b(y)$ into one logarithm.

Answer: $\log_b\left(\frac{x^2}{y^3}\right)$. Apply power rule backwards, then quotient rule backwards.

Flashcard 76: Evaluate: $\text{log}_5(125)$.

Answer: $3$. Since $5^3 = 125$, the logarithm equals 3.

Flashcard 77: What is $log_3\left(\frac{1}{9}\right)$?

Answer: $-2$. Since $3^{-2} = \frac{1}{9}$, we have $\log_3\left(\frac{1}{9}\right) = -2$.

Flashcard 78: For $0<b<1$, is $f(x)=\log_b(x)$ increasing or decreasing on $x>0$?

Answer: Decreasing on $x>0$. For bases between 0 and 1, larger inputs yield smaller logarithm values.

Flashcard 79: What restriction on the base $b$ is required for $log_b(x)$ to be a logarithm?

Answer: $b>0$ and $b\neq 1$. Base must be positive and not equal to 1 for logarithm to be well-defined.

Flashcard 80: Simplify using properties: $\text{log}_b(b^{-3})$.

Answer: $-3$. The logarithm and exponential with same base cancel.

Flashcard 81: Simplify $log_5(125x)$ as a sum of logs.

Answer: $\log_5(125x)=3+\log_5(x)$. Use product rule: $\log_5(125) + \log_5(x) = 3 + \log_5(x)$ since $5^3 = 125$.

Flashcard 82: Simplify $log_2(16x^3)$ using log rules.

Answer: $\log_2(16x^3)=4+3\log_2(x)$. Use product and power rules: $\log_2(16) + \log_2(x^3) = 4 + 3\log_2(x)$.

Flashcard 83: Find $\text{log}_{10}(1000)$.

Answer: $3$. Since $10^3 = 1000$, the common logarithm equals 3.

Flashcard 84: Simplify $log_5(125x)$ as a sum of logs.

Answer: $\log_5(125x)=3+\log_5(x)$. Use product rule: $\log_5(125) + \log_5(x) = 3 + \log_5(x)$ since $5^3 = 125$.

Flashcard 85: What is the change-of-base formula for $\log_b(x)$ using base $10$?

Answer: $\log_b(x)=\frac{\log(x)}{\log(b)}$. Converts any base logarithm to common logarithm (base 10).

Flashcard 86: What restriction on the input $x$ is required for $log_b(x)$ to be defined over the reals?

Answer: $x>0$. Input must be positive since logarithms of non-positive numbers are undefined in reals.

Flashcard 87: What is the inverse relationship between $b^x$ and $log_b(x)$?

Answer: $b^{\log_b(x)}=x$ and $\log_b(b^x)=x$. Exponential and logarithm functions are inverse operations that cancel each other.

Flashcard 88: What restriction on the base $b$ is required for $log_b(x)$ to be a logarithm?

Answer: $b>0$ and $b\neq 1$. Base must be positive and not equal to 1 for logarithm to be well-defined.

Flashcard 89: What is $log_b(b^k)$ when $b>0$ and $b\neq 1$?

Answer: $\log_b(b^k)=k$. Logarithm of a power of the base equals the exponent.

Flashcard 90: What is the definition of $log_b(x)$ in terms of an exponential equation?

Answer: $\log_b(x)=y\iff b^y=x$. Converts between logarithmic and exponential forms using equivalent definitions.

Flashcard 91: Solve for $x$: $log_2(x)+\log_2(x)=6$.

Answer: $x=8$. Simplify to $\log_2(x^2) = 6$, so $x^2 = 2^6 = 64$, giving $x = 8$.

Flashcard 92: What is the meaning of $log(x)$ on the ACT when no base is written?

Answer: $\log(x)=\log_{10}(x)$. When no base is specified, logarithm assumes base 10 (common logarithm).

Flashcard 93: For $b>1$, is $f(x)=\log_b(x)$ increasing or decreasing on $x>0$?

Answer: Increasing on $x>0$. For bases greater than 1, larger inputs yield larger logarithm values.

Flashcard 94: Evaluate: $\text{ln}(e)$.

Answer: $1$. Natural logarithm of its base equals 1.

Flashcard 95: What is the identity value of $\log_b(1)$ for any valid base $b$?

Answer: $\log_b(1)=0$. Any base raised to power 0 equals 1, so $\log_b(1) = 0$.

Flashcard 96: What is the quotient rule for logarithms: $log_b\left(\frac{M}{N}\right)$ equals what?

Answer: $\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)$. Quotient inside log equals difference of individual logs.

Flashcard 97: Evaluate: $\text{log}_2(32)$.

Answer: $5$. Since $2^5 = 32$, the logarithm equals 5.

Flashcard 98: What is $log_{10}(1000)$?

Answer: $3$. Since $10^3 = 1000$, we have $\log_{10}(1000) = 3$.

Flashcard 99: Find $\text{log}_{10}(1000)$.

Answer: $3$. Since $10^3 = 1000$, the common logarithm equals 3.

Flashcard 100: What is $log_2(8)$?

Answer: $3$. Since $2^3 = 8$, we have $\log_2(8) = 3$.