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When multiplying two powers that have the same base, you can add the exponents: $a^n \cdot a^m = a^{n+m}$.

When dividing two powers with the same base, subtract the exponents: $\frac{a^n}{a^m} = a^{n-m}$ where $a\neq0$.

Any nonzero number raised to the power of zero equals one: $a^0 = 1$ where $a\neq0$.

A negative exponent represents the reciprocal of the number raised to the opposite positive exponent: $a^{-n} = \frac{1}{a^n}$ where $a\neq0$.

A product raised to an exponent means each factor is raised to the exponent: $(ab)^n = a^n \cdot b^n$.

A quotient raised to an exponent means both numerator and denominator are raised to the exponent: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ where $b\neq0$.

When raising a power to a power, multiply the exponents: $(a^n)^m = a^{n \cdot m}$.

Multiple powers with different bases raised to the same power can be simplified: $(a\cdot b)^n = a^n \cdot b^n$.

The logarithm of a number with one base can be converted to a logarithm with a different base: $\log_b a = \frac{\log_c a}{\log_c b}$.

The logarithm of a product is the sum of the logarithms of the factors: $\log_b(a \cdot c) = \log_b(a) + \log_b(c)$.

The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator: $\log_b\left(\frac{a}{c}\right) = \log_b(a) - \log_b(c)$.

The logarithm of a number raised to an exponent is the exponent times the logarithm of the number: $\log_b(a^n) = n \cdot \log_b(a)$.

The natural logarithm, denoted as $\ln$, is the logarithm to the base $e$, where $e$ is the Euler's number approximately 2.71828: $\ln(x) = \log_e(x)$.

The logarithm of one, regardless of the base, is always zero: $\log_b(1) = 0$.

The logarithm of the base to itself is always one: $\log_b(b) = 1$.

Exponential and logarithmic functions are inverses of each other: $b^{\log_b(a)} = a$ and $\log_b(b^a) = a$.

The common logarithm is the logarithm with base 10, often denoted as $\log$: $\log(x) = \log_{10}(x)$.