how to use logarithms on calculator

What logarithms mean (in plain language)

A logarithm answers one question: “What exponent gives this number?”

For example, if you see log₁₀(1000), you are really asking: “10 to what power equals 1000?” Since 10³ = 1000, the answer is 3.

This is why logarithms and exponents are inverse operations. If exponentials build numbers up, logs peel them back down to the power.

Where the log buttons are on a calculator

LOG button

The LOG key is usually base 10. So LOG(100) returns 2, because 10² = 100.

LN button

The LN key is the natural logarithm (base e, where e ≈ 2.71828). LN(e) = 1, LN(1) = 0, and LN grows slowly as numbers get larger.

10^x and e^x buttons

These are inverse keys for LOG and LN:

• 10^x undoes LOG
• e^x undoes LN

If your calculator has a “2nd” or “SHIFT” button, press that first to access these inverse functions on many models.

How to use logarithms on calculator: step-by-step

Example 1: Calculate log₁₀(500)

• Type: 500
• Press: LOG
• Read result: about 2.69897

Interpretation: 10^2.69897 ≈ 500.

Example 2: Calculate ln(12)

• Type: 12
• Press: LN
• Result: about 2.4849

Interpretation: e^2.4849 ≈ 12.

Example 3: Calculate log base 2 of 50

Most calculators don’t have a dedicated log₂ button, so use the change-of-base formula:

log_b(x) = ln(x) / ln(b) (or LOG version works too)

• Compute LN(50)
• Compute LN(2)
• Divide: LN(50) ÷ LN(2)

Result is about 5.6439.

Common calculator mistakes (and fixes)

• Trying log of zero or a negative number: undefined in real-number mode. For standard classes, input must be x > 0.
• Mixing LOG and LN accidentally: LOG is base 10, LN is base e.
• Forgetting parentheses: use proper grouping on expressions like LN(3x + 2).
• Wrong mode confusion: log functions are not degree/radian dependent, but other parts of your expression might be.

When to use LOG vs LN

Use LOG (base 10) when the problem is written in powers of 10, pH, decibels, or common-log tables. Use LN when dealing with continuous growth/decay, calculus, and formulas based on e.

In practice, either can compute arbitrary bases with change-of-base, as long as you stay consistent in numerator and denominator.

Solving exponential equations with your calculator

Equation: 3^x = 20

Take logs on both sides:

x = log(20) / log(3)

Now calculate with your calculator. You’ll get x ≈ 2.7268.

Equation: e^2x = 7

Take natural log:

2x = ln(7), so x = ln(7) / 2 ≈ 0.973.

Quick reference

• LOG(x) = log₁₀(x)
• LN(x) = log_e(x)
• log_b(x) = ln(x) / ln(b)
• If y = log_b(x), then x = b^y

Final tip

After every logarithm result, do a quick inverse check:

• If you used LOG, test with 10^answer.
• If you used LN, test with e^answer.
• If you used base b, test with b^answer.

This 5-second habit catches most input mistakes and builds confidence fast.