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How do you solve an exponential equation of the form $a^x = b$?
Take the logarithm of both sides with base a: $x = \log_a(b)$. Or, take the natural logarithm: $x = \frac{\ln(b)}{\ln(a)}$
How do you solve a logarithmic equation of the form $\log_b(x) = a$?
Rewrite the equation in exponential form: $x = b^a$.
Steps to solve: $\log_b(x+2) + \log_b(x) = c$?
1. Use product rule: $\log_b((x+2)x) = c$. 2. Convert to exponential form: $(x+2)x = b^c$. 3. Solve the quadratic equation. 4. Check for extraneous solutions.
Steps to solve: $a^{f(x)} = a^{g(x)}$
1. Since the bases are equal, set the exponents equal to each other: $f(x) = g(x)$. 2. Solve for x. 3. Check for extraneous solutions.
Steps to solve: $a^{f(x)} = b$?
1. Take the logarithm of both sides (either base a or natural log). 2. Solve for x. 3. Check for extraneous solutions.
Steps to find the inverse of $y = 2e^{x+1} - 3$?
1. Swap x and y: $x = 2e^{y+1} - 3$. 2. Isolate the exponential term: $(x+3)/2 = e^{y+1}$. 3. Take the natural log: $\ln((x+3)/2) = y+1$. 4. Solve for y: $y = \ln((x+3)/2) - 1$.
Steps to find the inverse of $y = \log_3(x-2) + 1$?
1. Swap x and y: $x = \log_3(y-2) + 1$. 2. Isolate the log term: $x-1 = \log_3(y-2)$. 3. Convert to exponential form: $3^{x-1} = y-2$. 4. Solve for y: $y = 3^{x-1} + 2$.
Explain the inverse relationship between exponential and logarithmic functions.
Logarithmic functions 'undo' exponential functions, and vice versa. If $y = b^x$, then $x = \log_b(y)$.
Why is it important to check for extraneous solutions when solving logarithmic equations?
Logarithms are only defined for positive arguments. Solutions must be checked to ensure they don't result in taking the logarithm of a non-positive number.
Describe how transformations affect the graph of an exponential function.
The function $f(x) = ab^{x-h} + k$ shifts the basic exponential function horizontally by h, vertically by k, and stretches/compresses it by a factor of a.
Describe how transformations affect the graph of a logarithmic function.
The function $f(x) = a \log_b(x-h) + k$ shifts the basic logarithmic function horizontally by h, vertically by k, and stretches/compresses it by a factor of a.
How do you find the inverse of a function?
Swap x and y in the equation, then solve for y. This new equation represents the inverse function.
What is an extraneous solution?
A solution that emerges from the process of solving a problem but is not a valid solution to the original problem.
Define the inverse of a function.
A function that reverses the effect of another function. If f(a) = b, then f⁻¹(b) = a.
What is a logarithmic function?
The inverse function of an exponential function, expressing the power to which a base must be raised to produce a given number.
What is an exponential function?
A function in which the independent variable appears in the exponent.