Exponential Equation Solver
body {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
background-color: #f8f9fa;
color: #333;
line-height: 1.6;
margin: 0;
padding: 20px;
}
.loan-calc-container {
max-width: 700px;
margin: 30px auto;
background-color: #ffffff;
padding: 30px;
border-radius: 8px;
box-shadow: 0 4px 15px rgba(0, 0, 0, 0.1);
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h1, h2 {
color: #004a99;
text-align: center;
margin-bottom: 25px;
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.input-group {
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display: flex;
flex-direction: column;
gap: 8px;
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.input-group label {
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.input-group input[type="number"],
.input-group input[type="text"] {
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border: 1px solid #ccc;
border-radius: 4px;
font-size: 1rem;
box-sizing: border-box; /* Ensures padding doesn't affect width */
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button {
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color: white;
border: none;
padding: 12px 25px;
border-radius: 4px;
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font-weight: 600;
cursor: pointer;
transition: background-color 0.3s ease;
display: block;
width: 100%;
margin-top: 10px;
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button:hover {
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#result {
margin-top: 30px;
padding: 20px;
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border-radius: 4px;
text-align: center;
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#result p {
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.explanation {
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.explanation h2 {
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.explanation p, .explanation ul {
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/* Responsive adjustments */
@media (max-width: 768px) {
.loan-calc-container {
margin: 20px auto;
padding: 20px;
}
button {
font-size: 1rem;
padding: 10px 20px;
}
#result p {
font-size: 1.1rem;
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@media (max-width: 480px) {
.loan-calc-container {
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}
h1 {
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}
h2 {
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}
Exponential Equation Solver
Solve equations of the form b^x = y or b^(ax+c) = y.
Understanding Exponential Equations
Exponential equations are mathematical equations where the variable appears in the exponent. They are fundamental in many areas of science, finance, and technology.
The Standard Form
A common form of an exponential equation is:
bexponent = y
b is the base, a positive number not equal to 1.exponent is the expression involving the variable (usually x).y is the result, a positive number.
Solving Simple Exponential Equations (bx = y)
To solve for x in the simplest case (where the exponent is just x), we use logarithms. The definition of a logarithm states that if bx = y, then x = logb(y).
In practical terms, this means x is the logarithm of y with base b. Most calculators and programming languages provide functions for natural logarithm (ln, base e) and common logarithm (log, base 10). We can use the change of base formula:
logb(y) = ln(y) / ln(b) or logb(y) = log(y) / log(b)
Solving More Complex Exponential Equations (bax+c = y)
When the exponent is more complex, like ax + c, the process is similar but involves an extra algebraic step:
- Isolate the exponential term:
bax+c = y
- Take the logarithm of both sides (using any base, usually natural log 'ln'):
ln(bax+c) = ln(y)
- Use the logarithm property
ln(Mp) = p * ln(M): (ax + c) * ln(b) = ln(y)
- Isolate the term containing the variable:
ax + c = ln(y) / ln(b)
- Solve the resulting linear equation for
x:
ax = (ln(y) / ln(b)) - cx = ((ln(y) / ln(b)) - c) / a
Calculator Logic
This calculator handles equations in the form bexponentTerm = y. It parses the exponentTerm to identify the coefficients a and c from a linear expression like ax + c (or variations like x, 2x, x+5, etc.). If the exponentTerm is simply 'x', it assumes a=1 and c=0.
Use Cases
- Growth and Decay Models: Calculating population changes, radioactive decay rates, or compound interest over time.
- Physics: Analyzing phenomena like cooling or signal attenuation.
- Computer Science: Understanding algorithm complexity (e.g., O(2n)).
- Chemistry: Modeling reaction rates.
Example
Solve 32x + 1 = 27
- Base (b) = 3
- Exponent Term =
2*x + 1 (here a=2, c=1)
- Result Value (y) = 27
Calculation:
x = ((ln(27) / ln(3)) - 1) / 2
x = ((3) - 1) / 2
x = 2 / 2
x = 1
Check: 3(2*1 + 1) = 33 = 27. Correct!
function solveExponentialEquation() {
var base = parseFloat(document.getElementById("base").value);
var exponentTerm = document.getElementById("exponentTerm").value.trim();
var resultValue = parseFloat(document.getElementById("resultValue").value);
var solutionDisplay = document.getElementById("solutionValue");
var notesDisplay = document.getElementById("notes");
var resultDiv = document.getElementById("result");
// Clear previous results
solutionDisplay.innerText = "";
notesDisplay.innerText = "";
resultDiv.style.display = "none";
// — Input Validation —
if (isNaN(base) || base <= 0 || base === 1) {
notesDisplay.innerText = "Error: Base must be a positive number other than 1.";
resultDiv.style.display = "block";
return;
}
if (isNaN(resultValue) || resultValue xIndex) { // Found '+'
var cPart = exponentTerm.substring(plusIndex + 1);
c = parseFloat(cPart);
if (isNaN(c)) {
notesDisplay.innerText = "Error: Invalid constant term after '+'.";
resultDiv.style.display = "block";
return;
}
} else if (minusIndex > xIndex) { // Found '-'
var cPart = exponentTerm.substring(minusIndex + 1);
c = -parseFloat(cPart); // The sign is already handled by minusIndex
if (isNaN(c)) {
notesDisplay.innerText = "Error: Invalid constant term after '-'.";
resultDiv.style.display = "block";
return;
}
} else {
// No constant term found after 'x'
c = 0;
}
// Check if there are other characters besides ax+c
var remaining = exponentTerm.replace(axPart.replace('+', ") + 'x', ").replace( (plusIndex > xIndex ? '+' + cPart : (minusIndex > xIndex ? '-' + cPart : ") ) , ");
if(remaining !== "" && remaining !== "+" && remaining !== "-") {
notesDisplay.innerText = "Error: Invalid characters in exponent term.";
resultDiv.style.display = "block";
return;
}
} else {
// Exponent term does not contain 'x' – this is not solvable for 'x' in the exponent
notesDisplay.innerText = "Error: Exponent term must contain 'x' to solve for it.";
resultDiv.style.display = "block";
return;
}
// — Calculation —
var logBaseY = Math.log(resultValue); // ln(y)
var logBaseB = Math.log(base); // ln(b)
// Check for division by zero (shouldn't happen with base validation, but good practice)
if (logBaseB === 0) {
notesDisplay.innerText = "Error: Logarithm of base is zero (invalid base).";
resultDiv.style.display = "block";
return;
}
var exponentValue = logBaseY / logBaseB; // ln(y) / ln(b) = log_b(y)
var x;
if (a === 0) { // Should not happen if xExists is true, but for safety
notesDisplay.innerText = "Error: Coefficient 'a' is zero, cannot solve for x.";
resultDiv.style.display = "block";
return;
}
// Solve for x: a*x + c = exponentValue => a*x = exponentValue – c => x = (exponentValue – c) / a
x = (exponentValue – c) / a;
// — Display Result —
solutionDisplay.innerText = "x = " + x.toFixed(6); // Display with reasonable precision
notesDisplay.innerText = "Equation solved: " + base + "^(" + exponentTerm + ") = " + resultValue;
resultDiv.style.display = "block";
}