Identities and transformations

Identities are equations that are always true for all values of the variables involved. In the context of exponential functions and logarithms, identities help us simplify expressions and solve equations. Transformations refer to changing the form of an expression using these identities, such as rewriting an exponential expression as a logarithmic one, or vice versa.

Important properties

• Exponential identities: $a^{x+y} = a^x \cdot a^y$, $a^{x-y} = \frac{a^x}{a^y}$, $(a^x)^k = a^{xk}$

• Logarithmic identities: $\log_a(xy) = \log_a x + \log_a y$, $\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$, $\log_a(x^k) = k \log_a x$

• Change of base formula: $\log_a x = \frac{\log_b x}{\log_b a}$

• Exponential and logarithmic functions are inverses: $a^{\log_a x} = x$ and $\log_a(a^x) = x$