Log Calculator

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Log Calculator
Common Logarithm (Base 10)Natural Logarithm (Base e)Binary Logarithm (Base 2)Custom Base Logarithm
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function updateInputs(){var type=document.getElementById('log_type').value;var baseRow=document.getElementById('baseRow');if(type==='custom'){baseRow.style.display='table-row';}else{baseRow.style.display='none';}}function calculateLog(){var x=parseFloat(document.getElementById('input_x').value);var type=document.getElementById('log_type').value;var base;var result;var showSteps=document.getElementById('steps').checked;var solutionText="";if(isNaN(x)||x<=0){alert('The number (x) must be a positive value greater than 0.');return;}if(type==='common'){base=10;result=Math.log10(x);solutionText="log10("+x+") = "+result;}else if(type==='natural'){base=Math.E;result=Math.log(x);solutionText="ln("+x+") = "+result;}else if(type==='binary'){base=2;result=Math.log2(x);solutionText="log2("+x+") = "+result;}else{base=parseFloat(document.getElementById('input_base').value);if(isNaN(base)||base<=0||base===1){alert('The base (b) must be a positive value greater than 0 and not equal to 1.');return;}result=Math.log(x)/Math.log(base);solutionText="log"+base+"("+x+") = "+result;}var output="
Result: "+result.toFixed(6)+"
";if(showSteps){output+="
";output+="Step-by-Step Solution:
";output+="1. Identification: logb(x) = y
";output+="2. Values: b = "+base+", x = "+x+"
";output+="3. Change of base formula: log"+base+"("+x+") = ln("+x+") / ln("+base+")
";output+="4. Calculation: "+Math.log(x).toFixed(6)+" / "+Math.log(base).toFixed(6)+"
";output+="5. y = "+result.toFixed(6)+"
";output+="6. Verification: "+base+""+result.toFixed(4)+" ≈ "+x+"
";}document.getElementById('answer').innerHTML=output;}updateInputs();

How to Use the Log Calculator

The log calculator is a versatile tool designed to solve logarithmic equations for any given base. Logarithms are the inverse operation of exponentiation, essentially asking the question: "To what power must we raise the base to get this number?" This calculator simplifies complex math for students, engineers, and data scientists.

Logarithm Type
Choose between Common Log (Base 10), Natural Log (Base e), Binary Log (Base 2), or a Custom Base for your calculation.
Number (x)
The positive value you want to find the logarithm of. Note: Logarithms of zero or negative numbers are undefined in the real number system.
Base (b)
The base of the logarithm. If you select a custom base, it must be a positive number and cannot be equal to 1.

The Logarithm Formula

The fundamental definition of a logarithm is the relationship between a base, an exponent, and a result. If by = x, then:

logb(x) = y

To calculate a logarithm with an unusual base, we use the Change of Base Formula:

logb(x) = logk(x) / logk(b)

  • b: The base of the log.
  • x: The value we are calculating for.
  • k: A new base (usually base 10 or base e) that is available on standard calculators.
  • y: The resulting exponent.

Calculation Examples

Example 1: Finding a Common Logarithm

Suppose you want to find the value of log10(1000). Since 10 × 10 × 10 = 1000, we know that 103 = 1000. Therefore, the result is 3.

Example 2: Custom Base Calculation

Calculate log2(50) using the change of base formula with natural logs (ln):

  1. Set x = 50 and b = 2
  2. Apply formula: ln(50) / ln(2)
  3. Calculate ln(50) &approx; 3.912023
  4. Calculate ln(2) &approx; 0.693147
  5. Divide: 3.912023 / 0.693147 &approx; 5.6438
  6. Result = 5.6438

Common Questions

What is the "natural log"?

The natural log, denoted as ln(x), is a logarithm with the base e (Euler's number, approximately 2.71828). It is widely used in physics, finance, and statistics because it naturally describes growth and decay processes.

Can the base of a log be negative?

In standard real-number mathematics, the base must be positive and not equal to 1. Negative bases lead to complex numbers and are generally not supported by standard log calculators as they do not result in a continuous real-valued function.

Why can't I take the log of 0?

There is no power you can raise a positive base to that will result in 0. As the exponent becomes more negative, the result approaches 0 but never reaches it. Therefore, log(0) is undefined (approaching negative infinity).

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