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.
Understanding Math: Solving Exponential and Log Equations
When an unknown appears in an exponent, first try to express both sides with the same base. This works because equal powers of the same valid base have equal exponents. For example, if a problem can be rewritten as two raised to an unknown power equals two raised to the seventh power, the unknown is seven.
Many equations need a small amount of algebra before this step. Isolate the exponential expression by undoing addition, subtraction, multiplication, or division around it.
If matching bases is not possible, take a logarithm of both positive sides. The logarithm turns the exponent into a multiplier, making the unknown accessible.
Logarithmic equations need extra care because their inputs have restrictions. The expression inside every logarithm must be positive. A value of zero is not allowed, and neither is a negative value.
For instance, if the logarithm of a quantity minus two equals the logarithm of seven, the quantity must be greater than two before any solving begins. Since logarithms with the same base give the same output only when their positive inputs match, the quantity minus two equals seven. The candidate is nine, which passes the restriction.
Always substitute a final answer into the original equation. Algebra can produce a number that looks correct after simplification but makes a logarithm undefined.
The log properties are useful tools, but they are easy to misuse. A logarithm of a product can be separated into a sum of logarithms. A logarithm of a quotient can become a difference.
A power can move in front as a multiplier. These rules do not apply to addition inside a logarithm. The logarithm of a sum cannot be split into separate logarithms.
This is one of the most common errors. In a model such as five times one point zero three raised to time equals eight hundred, divide by five first.
Then take the natural logarithm of both sides. The exponent becomes time times the natural logarithm of one point zero three, so division isolates time.
These equations appear when a change happens by the same factor during each equal time interval. Compound interest uses a growth factor. Medicine in the body often uses a decay factor.
Acidity is measured with logarithms because hydrogen ion amounts can vary across a huge range. Sound levels and earthquake measurements use logarithmic scales for a similar reason. When using a calculator, use parentheses around a full numerator and denominator in a change of base calculation.
Keep extra decimal places until the final step, since repeated rounding can noticeably change a time or rate. Pay attention to units as well. A calculated time has meaning only when the growth or decay rate uses the same time unit.
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.