🔴 Exponential & Logarithmic Functions
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Properties of Logarithms

Apply product, quotient, and power rules to expand and condense logarithmic expressions.

The logarithm log_b(x) = y means b^y = x. Logarithms are the inverse of exponential functions. The three fundamental logarithm properties allow us to expand complex logarithmic expressions or combine multiple logs into one.

log⁡b(MN)=log⁡bM+log⁡bNlog⁡bMN=log⁡bM−log⁡bNlog⁡b(Mp)=plog⁡bM\log_b(MN) = \log_b M + \log_b N \qquad \log_b\frac{M}{N} = \log_b M - \log_b N \qquad \log_b(M^p) = p\log_b M
log⁡bb=1log⁡b1=0blog⁡bx=xlog⁡b(bx)=x\log_b b = 1 \qquad \log_b 1 = 0 \qquad b^{\log_b x} = x \qquad \log_b(b^x) = x
📝Expanding a Logarithm
Expand log₂(8x³/y²). = log₂(8) + log₂(x³) − log₂(y²) = 3 + 3log₂(x) − 2log₂(y).
log⁡28x3y2=log⁡28+3log⁡2x−2log⁡2y=3+3log⁡2x−2log⁡2y\log_2\frac{8x^3}{y^2} = \log_2 8 + 3\log_2 x - 2\log_2 y = 3 + 3\log_2 x - 2\log_2 y
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Remember This!

Change of base formula: log_b(x) = log(x)/log(b) = ln(x)/ln(b). This lets you evaluate any logarithm using a calculator that only has log (base 10) or ln (base e).

✏️ Try It!

Which expression is equivalent to log(x²y / z)?

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