Exponential and logarithmic equations appear whenever a quantity grows, decays, compounds, or changes by repeated multiplication. They are used in finance, biology, chemistry, physics, computer science, and data modeling. The main idea is that logarithms undo exponentials, just as subtraction undoes addition.
Learning to switch between exponential and logarithmic form gives you a powerful way to solve for unknown exponents.
Key Facts
- Exponential to logarithmic form: a^x = b means log_a(b) = x, where a > 0, a != 1, and b > 0.
- Common log and natural log: log(x) means log_10(x), while ln(x) means log_e(x).
- Power property: log_a(M^p) = p log_a(M), which helps bring an exponent down where you can solve for it.
- Product property: log_a(MN) = log_a(M) + log_a(N), for M > 0 and N > 0.
- Quotient property: log_a(M/N) = log_a(M) - log_a(N), for M > 0 and N > 0.
- Change of base: log_a(b) = ln(b)/ln(a) = log(b)/log(a), which lets you evaluate logs on a calculator.
Vocabulary
- Exponential equation
- An equation in which the unknown variable appears in an exponent, such as 3^x = 81.
- Logarithmic equation
- An equation that contains a logarithm of an expression involving a variable, such as log_2(x + 1) = 4.
- Base
- The base is the repeated factor in an exponential expression or the number that defines a logarithm, such as 5 in 5^x or log_5(x).
- Domain
- The domain is the set of input values that make an expression valid, especially requiring log arguments to be positive.
- Inverse functions
- Inverse functions undo each other, so y = a^x and x = log_a(y) reverse the same relationship.
Common Mistakes to Avoid
- Taking the log of only one side is wrong because any operation used to solve an equation must be applied to both sides equally.
- Forgetting the log domain is wrong because log(x), log(x - 3), and ln(2x + 1) require their arguments to be greater than 0 before a solution can be accepted.
- Writing log(M + N) = log(M) + log(N) is wrong because logarithm properties work for products and quotients, not sums or differences.
- Dropping parentheses after using the power property is wrong because log((x + 2)^3) = 3 log(x + 2), not 3 log x + 2.
Practice Questions
- 1 Solve for x: 5^(x - 1) = 125.
- 2 Solve for x and check the domain: log_3(x + 2) + log_3(x - 2) = 2.
- 3 A student solves ln(x - 4) = ln(2x - 9) and gets x = 5. Explain why checking the domain matters and decide whether the solution is valid.