Exponentials and Logarithms

Exponential Equations

• Definition: Equations in which a variable appears in the exponent.

• P(r+r/n)^nt represents the formula for compound interest, where P is the principal amount, r is the annual interest rate, n is the number of times that interest is compounded per year, and t is the number of years the money is invested or borrowed.

• Pe^rt represents the continuous growth of an investment or population over time, where e is the base of the natural logarithm, approximately equal to 2.71828.

  Logarithms

• are the inverses of exponential functions, allowing us to solve for the exponent in equations of the form b^y = x, where b is the base, y is the exponent, and x is the result.

• Logarithms can be divided into four pieces, Log, base 10, base e, and ln.

• Logarithm (Log): The logarithm to the base 10, commonly used in scientific calculations.

  
  

• Natural Logarithm (ln): The logarithm to the base e, crucial in calculus and continuous growth models.

• Change of Base Formula: This allows the conversion between different logarithm bases, expressed as log_b(x) = log_k(x) / log_k(b) for any base k.

• Properties of Logarithms: Key properties include the product, quotient, and power rules, which facilitate the simplification of logarithmic expressions.

How to solve

Logarithms are usually solved by condensing, expanding, or solving.

Expanding a Logarithm is the process of rewriting a logarithm. to do it we must apply the relevant properties, such as the product rule to separate products into sums, the quotient rule to transform divisions into differences, and the power rule to bring down exponents as coefficients.

To condense a logarithm, we must use the inverse of the properties applied during expansion; specifically, we combine sums into products, differences into divisions, and coefficients back into exponents.