Finding the Exponent That Produces a Value
A logarithm answers a specific question: which exponent produces a given value? In that sense, logarithmic and exponential functions undo each other.
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Start with this exponential equation:
The exponent that turns into is . Logarithmic notation records the same relationship from the opposite direction:
In general, exponential and logarithmic notation are related by:
A logarithmic function with base (where and ) is expressed as:
| Name | Base | Notation and example | Reason |
|---|---|---|---|
| Common logarithm | |||
| Natural logarithm | |||
| Binary logarithm |
Exchanging coordinates explains the reflection in the graph. The point on the exponential graph becomes on the logarithmic graph. Every logarithmic input remains positive.
Graph characteristics with :
Each property follows from the definition of the logarithm as the inverse of the exponential function, because undoing the exponential function must undo its rules too. The basic properties concern a single value, and the operational properties concern a product, a quotient, or a power.
The first two identities read exponents zero and one. The next two undo logarithms and exponentials with the same base.
The rules below require positive arguments: . Both bases satisfy and , and can be any real number.
For example, multiplication inside a logarithm becomes addition of two logarithmic values.
Indonesia's Ministry of Health, or Kemenkes, reported two confirmed COVID-19 cases on 2 March 2020.
On 30 April 2020, the ministry reported 10,118 cumulative confirmed cases.
The dates are apart.
To practise logarithms, we can fit a simple exponential curve through those two reported values. Let be the number of elapsed days after 2 March and the fitted case count. Since and , the model is:
To find the rate fitted to the two data points, start with the following model. The initial report determines the multiplier, and the day-59 report determines the rate.
If the same fitted rate were extended, logarithms would place the threshold at:
This calculation uses logarithms to solve the exponential model. Real confirmed case counts also depend on testing, reporting, behaviour, and public-health measures. A fixed exponential rate is useful only while its assumptions still fit the situation.
These four problems move from reading a single logarithm to solving an exponential model. Work each one before the worked solutions, then check your answers against the steps that follow.
Determine the value of , , and .
If , determine the value of .
Simplify:
An investment grows according to the formula (in million rupiah). How many years are needed for the investment to become ?
Problem 1. Read each logarithm as an exponent question. The required powers are:
Problem 2. Rewrite the logarithmic equation in exponential form:
Substituting gives , so the result checks out.
Problem 3. Evaluate the three powers of before combining them:
Problem 4. The formula measures in millions of rupiah, so . Substitute the target and isolate the exponential term:
The investment reaches the target after .
Published: . Updated: .