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  2. AP Pre Calculus
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What is the product rule for logarithms?

log⁡b(MN)=log⁡b(M)+log⁡b(N)\log_b(MN) = \log_b(M) + \log_b(N)logb​(MN)=logb​(M)+logb​(N)

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What is the product rule for logarithms?

log⁡b(MN)=log⁡b(M)+log⁡b(N)\log_b(MN) = \log_b(M) + \log_b(N)logb​(MN)=logb​(M)+logb​(N)

What is the quotient rule for logarithms?

log⁡b(M/N)=log⁡b(M)−log⁡b(N)\log_b(M/N) = \log_b(M) - \log_b(N)logb​(M/N)=logb​(M)−logb​(N)

What is the power rule for logarithms?

log⁡b(Mp)=p⋅log⁡b(M)\log_b(M^p) = p \cdot \log_b(M)logb​(Mp)=p⋅logb​(M)

What is the change of base formula for logarithms?

log⁡b(M)=log⁡c(M)log⁡c(b)\log_b(M) = \frac{\log_c(M)}{\log_c(b)}logb​(M)=logc​(b)logc​(M)​

General form of exponential function with transformations

f(x)=abx−h+kf(x) = ab^{x-h} + kf(x)=abx−h+k

General form of logarithmic function with transformations

f(x)=alog⁡b(x−h)+kf(x) = a \log_b(x-h) + kf(x)=alogb​(x−h)+k

Inverse of exponential function f(x)=abx−h+kf(x) = ab^{x-h} + kf(x)=abx−h+k

f−1(x)=h+ln⁡((x−k)/a)ln⁡(b)f^{-1}(x) = h + \frac{\ln((x-k)/a)}{\ln(b)}f−1(x)=h+ln(b)ln((x−k)/a)​

Inverse of logarithmic function f(x)=alog⁡b(x−h)+kf(x) = a \log_b(x-h) + kf(x)=alogb​(x−h)+k

f−1(x)=b(x−k)a+hf^{-1}(x) = b^{\frac{(x-k)}{a}} + hf−1(x)=ba(x−k)​+h

What are the key differences between solving exponential equations and logarithmic equations?

Exponential: Isolate exponential term, take logarithm. Logarithmic: Isolate log term, convert to exponential form. Extraneous solutions are more common in logarithmic equations.

Compare the domain restrictions of exponential and logarithmic functions.

Exponential: Domain is all real numbers. Logarithmic: Domain is positive real numbers (argument must be > 0).

Explain the inverse relationship between exponential and logarithmic functions.

Logarithmic functions 'undo' exponential functions, and vice versa. If y=bxy = b^xy=bx, then x=log⁡b(y)x = \log_b(y)x=logb​(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)=abx−h+kf(x) = ab^{x-h} + kf(x)=abx−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)=alog⁡b(x−h)+kf(x) = a \log_b(x-h) + kf(x)=alogb​(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.