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Express The Following As A Single Logarithm Calculator

Reviewed by Calculator Editorial Team

This calculator helps you combine multiple logarithms into a single logarithm using the logarithm rules. Whether you're solving math problems, simplifying expressions, or working with scientific calculations, this tool provides a clear, step-by-step solution.

How to Use This Calculator

To express multiple logarithms as a single logarithm, follow these steps:

  1. Enter the first logarithm in the format logb(x).
  2. Select the operation (+, -, ×, ÷) to combine with the next logarithm.
  3. Enter the second logarithm in the same format.
  4. Click "Calculate" to see the simplified single logarithm.
  5. Review the step-by-step solution and the final result.

The calculator applies logarithm rules to combine the expressions. You can also use the calculator to verify your manual calculations or explore different combinations.

Logarithm Rules

The calculator uses these fundamental logarithm rules to combine expressions:

logb(x) + logb(y) = logb(x × y) logb(x) - logb(y) = logb(x ÷ y) logb(xy) = y × logb(x) logb(x) = logb(y) ÷ logb(x) (Change of base formula)

These rules allow you to combine logarithms with the same base. If the bases differ, you can first convert them to the same base using the change of base formula.

Worked Examples

Example 1: Adding Logarithms

Combine log2(5) + log2(7) using the calculator:

  1. Enter log2(5) as the first logarithm.
  2. Select the + operation.
  3. Enter log2(7) as the second logarithm.
  4. Click "Calculate".

The calculator shows the step-by-step solution: log2(5) + log2(7) = log2(5 × 7) = log2(35).

Example 2: Subtracting Logarithms

Combine log3(12) - log3(4) using the calculator:

  1. Enter log3(12) as the first logarithm.
  2. Select the - operation.
  3. Enter log3(4) as the second logarithm.
  4. Click "Calculate".

The calculator shows the step-by-step solution: log3(12) - log3(4) = log3(12 ÷ 4) = log3(3).

FAQ

Can I combine logarithms with different bases?

Yes, you can use the change of base formula to convert logarithms to the same base before combining them. The calculator handles this automatically when the bases differ.

What if the arguments of the logarithms are negative?

Logarithms of negative numbers are not defined in real numbers. The calculator will show an error if you enter a negative argument.

How accurate are the results?

The calculator uses standard logarithm rules and provides exact results. For decimal approximations, you can use a scientific calculator.