Logarithm Calculator & Solver

Evaluate logarithms of any base, convert between bases, and solve exponential equations. This logarithm calculator provides step-by-step solutions with a handy reference table of common values.

What Is a Logarithm?

A logarithm answers the question: “To what power must the base \(b\) be raised to produce \(x\)?” Formally, \(\log_b(x) = y\) means \(b^y = x\). For example, \(\log_2(8) = 3\) because \(2^3 = 8\). Logarithms are the inverse operation of exponentiation.

Common Logarithm Bases

Three bases appear most frequently in mathematics and science:

• Common logarithm (base 10): Written as \(\log(x)\) or \(\log_{10}(x)\). Used in the pH scale, decibels, and the Richter scale.
• Natural logarithm (base e): Written as \(\ln(x)\). The number \(e \approx 2.71828\) arises naturally in calculus, compound interest, and growth/decay models.
• Binary logarithm (base 2): Written as \(\log_2(x)\). Essential in computer science for analyzing algorithms, data structures, and information theory.

Logarithm Properties

The following properties hold for any valid base \(b > 0, b \neq 1\):

$$\log_b(xy) = \log_b(x) + \log_b(y) \quad \text{(Product Rule)}$$

$$\log_b\!\left(\frac{x}{y}\right) = \log_b(x) – \log_b(y) \quad \text{(Quotient Rule)}$$

$$\log_b(x^n) = n \cdot \log_b(x) \quad \text{(Power Rule)}$$

Change of Base Formula

To convert a logarithm from one base to another, use the change of base formula:

$$\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$$

This is particularly useful because most calculators only have buttons for \(\log_{10}\) and \(\ln\). For example, \(\log_2(10) = \frac{\ln(10)}{\ln(2)} \approx 3.3219\).

Solving Exponential Equations

Logarithms are the primary tool for solving equations where the unknown is in the exponent. To solve \(b^x = y\), take the logarithm of both sides:

$$x = \frac{\log(y)}{\log(b)} = \log_b(y)$$

This technique applies to problems in radioactive decay (\(N = N_0 e^{-\lambda t}\)), population growth, and compound interest calculations.

Antilogarithms

The antilogarithm (or inverse logarithm) reverses the logarithm operation. If \(\log_b(x) = y\), then the antilog is \(x = b^y\). For common logarithms, \(\text{antilog}(2) = 10^2 = 100\). Antilogs are used to convert values from logarithmic scales (like decibels or pH) back to linear scales.

Real-World Applications of Logarithms

Logarithms appear throughout science and everyday life. The Richter scale measures earthquake intensity logarithmically — each whole number increase represents a tenfold increase in amplitude. Sound intensity is measured in decibels (dB), a logarithmic scale. In finance, logarithms help calculate the time needed for an investment to reach a target value with compound interest. In biology, the pH scale is a negative logarithm of hydrogen ion concentration.

A logarithm answers an exponent question. If \(2^5=32\), then \(\log_2 32=5\). Keeping that sentence in view prevents most sign and base mistakes.

How to use this calculator

Enter a positive argument and a base greater than zero but not equal to 1. Use base 10 for common logarithms, base \(e\) for natural logarithms, and another base when the exponential model specifies it.

When solving an equation, isolate the logarithm or exponential expression before converting forms. Then check the final value in the original equation, because algebraic steps can produce a value outside the logarithm’s domain.

1. Rewrite logarithmic form as exponential form when the unknown is an exponent.
2. Use product, quotient, and power rules to combine logarithms only when their bases match.
3. Apply the change-of-base formula when your calculator lacks the required base.
4. Check that every logarithm argument is positive.

Worked example

Solve \(\log_3(x-1)=4\). Convert to exponential form: \(x-1=3^4=81\). So \(x=82\). The domain check passes because \(x-1=81>0\).

For \(\log_2 x+\log_2(x-2)=3\), combine the logs to get \(\log_2[x(x-2)]=3\). Then \(x^2-2x=8\), giving candidates 4 and -2. Only 4 works because both original arguments must be positive.

How to read the result

The base controls how quickly powers grow. A base above 1 produces an increasing logarithm; a base between 0 and 1 produces a decreasing one. The argument tells you which positive output of the exponential function you are reversing.

Logarithms turn multiplication into addition and powers into products. That is why they appear in decibels, pH, earthquake magnitude, compound growth, and algorithm analysis.

Common mistakes to avoid

• Using \(\log(a+b)=\log a+\log b\). No such sum rule exists.
• Dropping the base and mixing log values from different bases.
• Keeping a solution that makes an original logarithm argument zero or negative.
• Confusing \(\log_b x\) with \(b/x\). A logarithm is an inverse exponent, not division.

How to verify the result

Reverse the logarithm to check the result. If \(\log_b x=y\), then \(b^y\) must reproduce x. For an equation, substitute every candidate into the original logarithms and reject any value that makes an argument zero or negative.

The base must be positive and cannot equal 1. Product, quotient, and power rules apply only to logarithms with compatible bases; there is no rule that turns \(\log(a+b)\) into \(\log a+\log b\).

Keep an exact change-of-base expression such as \(\ln 5/\ln 2\) through the algebra. Round the final decimal only when the problem asks for an approximation, because early rounding can move an exponent enough to fail substitution.

Limits of the calculation

This solver works in the real-number setting unless it explicitly reports complex values. Negative arguments require complex logarithms, which are multi-valued and need a branch convention.

Rounded decimal answers can hide an exact form such as \(\ln 5/\ln 2\). Keep the exact expression through the algebra and round only the final result.

Related calculators

Use pH Calculator, Compound Interest Calculator, Inverse Function Calculator when the next part of the problem needs a different method.

Useful algebra books and tools

For logarithms, practice moving between exponential and logarithmic form and checking the domain before accepting a solution. The best way to make an algebra calculator useful is to pair its result with enough written practice to recognize the underlying pattern.

• Practice the method: Browse algebra textbooks and workbooks on Amazon. Choose a book with worked solutions and mixed problem sets, not only formula summaries.
• Check routine calculations: Browse scientific calculators on Amazon. Use one to verify arithmetic after you have written the algebraic steps yourself.

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