Introduction
A logarithm asks a focused question: to what power must a given base be raised to produce a given positive number? Exponential notation states the power relationship directly, while logarithmic notation names the exponent. Converting between the two forms lets us read the same relationship in either direction.
For example, the statement \(2^3=8\) says that raising 2 to the third power gives 8. In logarithmic form, it says that the logarithm of 8 with base 2 is 3. The two expressions contain the same three pieces of information: the base, the exponent, and the resulting value.
The Definition and the Conversion Rule
For a real logarithm, the base must be positive and different from 1, and the argument must be positive. The exponent may be any real number. These conditions ensure that the exponential function with that base has an inverse on the positive real numbers.
This equivalence is the conversion rule. To move from logarithmic to exponential form, keep the base \(b\), make the logarithm’s argument \(x\) the result of a power, and use the logarithm’s value \(y\) as the exponent. To move the other way, identify the base, exponent, and result in \(b^y=x\), then write \(\log_b(x)=y\).
The positions of the symbols matter. In \(\log_b(x)=y\), \(b\) is the base, \(x\) is the argument, and \(y\) is the exponent being asked for. In \(b^y=x\), those same quantities appear as base, exponent, and result, respectively.
Why the Two Forms Are Equivalent
For a fixed base \(b>0\), \(b\ne1\), the exponential function takes an exponent \(y\) to the positive number \(b^y\). The logarithm with base \(b\) is its inverse: it takes that positive output back to the exponent that produced it. This is the inverse-function relationship established in the earlier material on logarithms as inverse functions.
Suppose first that \(\log_b(x)=y\). By the meaning of the inverse function, applying the exponential function with base \(b\) to \(y\) returns \(x\). Thus \(b^y=x\). Conversely, if \(b^y=x\), then \(x\) is the output of the base-\(b\) exponential function at input \(y\). Applying its inverse, \(\log_b\), to both sides gives \(\log_b(x)=y\). Both directions hold under the stated conditions, so the two forms express exactly the same relationship.
A Reliable Translation Process
When converting, it helps to label the three quantities before writing the new form. This is especially useful when the equation contains a variable or a negative exponent.
In a logarithm it is written as a subscript; in an exponential expression it is the number being raised to a power.
In logarithmic form it is the value on the other side of the equals sign. In exponential form it is the superscript.
In logarithmic form it is the argument inside the parentheses. In exponential form it is the number on the other side of the equals sign.
Use base to the exponent equals the result, or reverse that arrangement to write the logarithm. Substitute the proposed value to verify the power relationship.
Worked Examples
Worked Example: Convert and evaluate a logarithm
Write \(\log_2(32)=5\) in exponential form, then check that the statement is true.
The base is \(2\), the argument is \(32\), and the stated exponent is \(5\). The conversion rule gives:
Evaluate the power by repeated multiplication:
The exponential equation is true, so the logarithmic statement is true as well. The check confirms that 5 is the exponent that produces 32 from base 2.
Worked Example: Convert a negative exponent to logarithmic form
Write \(5^{-2}=\frac{1}{25}\) in logarithmic form, and verify the conversion.
The base is \(5\), the exponent is \(-2\), and the result is \(\frac{1}{25}\). In logarithmic form, the result goes inside the logarithm and the exponent becomes the value:
To check the power, use the negative-exponent rule:
Thus the exponential equation is correct, and the logarithmic statement follows. A negative logarithm value is possible: it means the exponent is negative. It does not mean the logarithm’s argument is negative.
Worked Example: Find an exponent from a known power
Find \(y\) if \(4^y=64\), and express the relationship in logarithmic form.
The given equation is already in exponential form: the base is \(4\), the unknown exponent is \(y\), and the result is \(64\). Reversing the conversion gives:
Since \(4\cdot4\cdot4=16\cdot4=64\), we have \(4^3=64\). Therefore the exponent is \(y=3\). Check by substituting this value into the original equation:
The substitution reproduces the given result, so \(y=3\). In logarithmic form, the checked statement is \(\log_4(64)=3\).
Worked Example: Use a fractional exponent
Convert \(9^{1/2}=3\) to logarithmic form and check it.
The base is \(9\), the exponent is \(\frac12\), and the result is \(3\). Therefore:
The exponent \(\frac12\) denotes the square root, so \(9^{1/2}=\sqrt{9}=3\), using the positive square root. This verifies the exponential equation and hence the logarithmic statement. Notice that the argument of the logarithm is \(3\), while its value is \(\frac12\).
Conditions and Common Errors
The conversion rule does not remove the requirements for a real logarithm. The base must satisfy \(b>0\) and \(b\ne1\), and the argument must satisfy \(x>0\). These conditions concern the logarithmic expression, even if a written power equation might appear to make sense without them.
A common error is to think that a positive result from an exponential equation permits a negative argument in a logarithm. For instance, the incorrect claim \(\log_2(-8)=3\) may come from noticing that \(2^3=8\) and then changing the sign of the result. But conversion would require \(2^3=-8\), which is false: \(2^3=8\). Also, \(\log_2(-8)\) is not defined as a real logarithm because its argument is negative.
Another frequent mistake is to exchange the exponent and the result. From \(2^3=8\), the correct conversion is \(\log_2(8)=3\), not \(\log_2(3)=8\). A quick check is to ask what question the logarithm asks: \(\log_2(8)\) asks which exponent on 2 produces 8, and the answer is 3.
Why This Conversion Is Useful
Logarithmic and exponential forms emphasize different parts of the same relationship. Exponential form makes the power calculation visible. Logarithmic form identifies the exponent as the unknown or as the quantity of interest. Converting accurately is therefore a basic tool for reading logarithmic expressions and for checking numerical claims about them.
The conversion also provides a direct way to check an answer without relying on a calculator: if a proposed value for \(\log_b(x)\) is \(y\), compute or recognize \(b^y\). The answer is correct exactly when that power equals \(x\). For simple powers, this check is often enough to evaluate the logarithm exactly.
Keep the three roles distinct: the base is what is raised, the logarithm’s value is the exponent, and the argument is what the power produces. Once those roles are identified, translation between the two forms is a precise rearrangement of one relationship, not a new calculation rule.
Check Your Understanding
For each question, identify the base, exponent, and result before converting or evaluating.
- Rewrite \(\log_3(27)=3\) in exponential form and verify it.
- Write \(2^{-3}=\frac18\) in logarithmic form.
- If \(5^y=125\), write the equivalent logarithmic statement and find \(y\).
- Explain why \(\log_4(-16)\) is not defined as a real logarithm.
- Correct the conversion \(\log_7(49)=2 \Longleftrightarrow 7^2=49\) if any part of it is incorrect.
