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The Problem Solver's Resource
 Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 2.

Exponentials and Logarithms

This is just a quick review of logarithms and exponents; it's elementary content.

Definitions

• Exponentials: Do you really need this one?
• Logarithms: If $b^a=x$, $\log_b{x}=a$. Note that a logarithm in base e, i.e. $\log_e{x}=a$ is notated as $\ln{x}=a$, or the natural logarithm of x. If no base is specified, then a logarithm is assumed to be in base 10.

Rules of Exponentiation and Logarithms

$a^x \cdot a^y=a^{x+y}$

$(a^x)^y=a^{xy}$

$\frac{a^x}{a^y}$=a^{x-y}$$(Error compiling LaTeX. ! Missing  inserted.)a^0=1$, where$a\ne 0$.$\log_b{xy}=\log_b{x}+\log_b{y}$$ (Error compiling LaTeX. ! Missing $inserted.)\log_b{x^y}=y\cdot \log_b{x}$$(Error compiling LaTeX. ! Missing inserted.)\log_b{\frac{x}{y}}=\log_b{x}-\log_b{y}$$ (Error compiling LaTeX. ! Missing$ inserted.)\log_b{a}=\frac{1}{\log_a{b}}$$(Error compiling LaTeX. ! Missing  inserted.)\log_b{b}=1$$ (Error compiling LaTeX. ! Missing $inserted.)\log_b{a}=\frac{\log_x{a}}{\log_x{b}}$, where x is a constant.$\log_1{a}$and$\log_0{a}$ are undefined.