Functions a^x and log_b x

Understanding Functions a^x

Exponential functions are those of the form a^x, where a is a constant base and x is the exponent. For instance, functions like 2^x_, 3^x, e^x are all examples of exponential functions.
The domain of exponential functions is all real numbers and the range is all positive real numbers.
Exponential growth is when an amount is increasing very quickly due to the fact that it is continually being multiplied by a number greater than one, known as the base. If a > 1, then the function a^x is an exponential growth function.
Exponential decay is when an amount is decreasing rapidly due to the fact that it is continually being multiplied by a number less than one but more than zero, known as the base. If 0 < a < 1, then the function a^x is an exponential decay function.
These functions are continuous, meaning they don’t have any breaks or holes, and they are infinitely differentiable, meaning you can find their derivative as many times as you like.

Laws of Indices

When you multiply two powers with the same base, you add the exponents: a^m * a^n = a^(m+n).
When you divide two powers with the same base, you subtract the exponents: a^m / a^n = a^(m-n).
A power raised to another power means you multiply the exponents: (a^m)^n = a^(mn).
a^0 = 1 for any number a, as anything raised to the power of zero is one.
a^(-n) = 1/a^n is the rule for negative exponents, which means you take the reciprocal of the base.

Understanding Logarithmic Functions log_b x

Logarithmic functions are the inverse of exponential functions. If y = a^x, then x = log_a y.
Logarithms are a way of expressing exponential relationships in terms of multiplicative relationships.
Logarithms follow several important properties known as the Laws of Logarithms. These are analogous to the laws of indices and are a critical part of understanding and simplifying logarithmic expressions.
The “Product Law” states that the log of a product is the sum of the logs: log_b (mn) = log_b m + log_b n.
The “Quotient Law” states that the log of a quotient is the difference of the logs: log_b (m/n) = log_b m - log_b n.
The “Power Law” states that the log of a power is the power times the log: log_b m^n = n log_b m.
log_b b = 1 and log_b 1 = 0 are known as the basic properties of logarithms.

Change of Base Formula

The change of base formula allows you to compute the logarithm of a number in one base in terms of the logarithm of that number in another base. It is given by log_b a = log_c a / log_c b for any base c. Using this formula, you can solve for logarithms in bases where the exact value may not be known.

Remember, practice is key in mastering these concepts. You should solve a variety of problems on exponential and logarithmic functions to improve your understanding.