🧮 algebra

1. **State the problem:** We are given that $\log_a x = 2$, $\log_a y = 5$, and $\log_a z = -3$. We need to find the value of $\log_a\left(\frac{xy^3}{z^2}\right)$. 2. **Recall log

1. **State the problem:** We need to find the least common denominator (LCD) for the equation $$\frac{3}{x} - \frac{2}{x+2} = \frac{6}{2x+4}$$. 2. **Identify denominators:** The de

1. **State the problem:** Solve the logarithmic equation $$\log_{10}(2x) = 3$$. 2. **Recall the definition of logarithm:** $$\log_{10}(y) = z$$ means $$10^z = y$$.

1. **State the problem:** We need to determine which of the given equations represents a one-to-one function. 2. **Recall the definition:** A function is one-to-one if each output

1. **State the problem:** We need to find the value of $f(4) + g(4)$ given the functions $f(x) = 2x^2$ and $g(x) = -3x + 15$. 2. **Recall the formulas:**

1. **State the problem:** We need to find the value of the composite function $f(g(-2))$ given $f(x)=4-x^2$ and $g(x)=2x+5$. 2. **Recall the formula for composite functions:** $f(g

1. **State the problem:** Evaluate $$\log_6(6) + \log_6(36) + \log_6(216)$$ using properties of logarithms. 2. **Recall the logarithm property:** $$\log_b(x) + \log_b(y) = \log_b(x

1. **State the problem:** We need to find the value of $f(-2)g(-2)$ given the functions $f(x)=x^2+4$ and $g(x)=2x+1$. 2. **Recall the formulas:**

1. **State the problem:** Simplify the rational expression $$\frac{x^2 - 9x + 18}{x^2 - x - 30}$$ by factoring and canceling common factors. 2. **Factor numerator:**

1. **State the problem:** We need to write $\frac{1}{x} - \frac{1}{x - 2}$ as a single rational expression. 2. **Formula and rules:** To subtract fractions, find a common denominat

1. **State the problem:** Write the expression $\frac{1}{x^3} \div \frac{2}{x}$ as a single rational expression.

1. **State the problem:** We need to find the sum of the first 10 terms of the geometric sequence $$\{1.5, 3, 6, 12, 24, ...\}$$. 2. **Identify the first term and common ratio:** T

1. **State the problem:** We need to find the value of $t$ that satisfies the equation $$5(2^t) = 4.$$\n\n2. **Isolate the exponential term:** Divide both sides of the equation by

1. **State the problem:** We need to find which rational function has a vertical asymptote at $x = -2$. 2. **Recall the rule for vertical asymptotes:** Vertical asymptotes occur wh

1. **State the problem:** We need to find the domain of the function $f(x) = \log(x - 8)$. The domain is the set of all $x$ values for which the function is defined. 2. **Recall th

1. **State the problem:** We need to find the domain and range of the function $$y = \sqrt{4 - x}$$. 2. **Recall the domain rule for square roots:** The expression inside the squar

1. **State the problem:** We want to find how many years, $t$, it will take for the population to decrease from 8,500 to 5,500 given a 3% annual decrease. 2. **Formula used:** The

1. **State the problem:** We are given the function $$f(x) = -3x^2 + 4x + 10$$ and need to find the value of $$f(2)$$. 2. **Formula used:** To find $$f(2)$$, substitute $$x = 2$$ i

1. **State the problem:** Graph the function $$f(x) = \frac{3}{4} \left(\frac{3}{7}\right)^x$$ and answer questions about its behavior.

1. **State the problem:** We want to express $3 \cdot \log_5 4$ as a single logarithm. 2. **Recall the logarithm power rule:** The property $a \cdot \log_b c = \log_b (c^a)$ allows

1. **State the problem:** We need to find which rational expression has domain restrictions $x \neq 4$ and $x \neq -3$. 2. **Recall domain restrictions:** The domain restrictions f