Exponential Equations Revision Notes for Grade 10 NSC Matric Mathematics

Exponential Equations

What are exponential equations?

Exponential equations are mathematical equations where the unknown variable appears in the exponent. These equations require special techniques to solve because the variable is not in its usual position as a base or coefficient.

Examples of exponential equations include:

$3^{x+1} = 9$
$5^t + 3 \times 5^{t-1} = 400$

The fundamental rule for solving exponential equations

When solving exponential equations, the most important principle to remember is the same base rule:

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If $a > 0$ and $a \neq 1$ , then: $a^x = a^y \text{ implies } x = y$

This means that when both sides of an equation have the same base, you can equate the exponents directly. However, this rule only works when the base is positive and not equal to 1.

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Important note: If $a = 1$ , then $x$ and $y$ can have different values, since $1$ raised to any power always equals $1$ .

Method 1: Equating exponents with the same base

This is the most straightforward method when you can express both sides of the equation using the same base.

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Worked Example 1: Basic same base method

Question: Solve $3^{x+1} = 9$

Solution:

Express both sides using the same base: Since $9 = 3^2$ , we can write: $3^{x+1} = 3^2$
Apply the same base rule: Since both sides have base 3, we can equate the exponents: $x + 1 = 2$
Solve for the variable: $x = 1$

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Worked Example 2: Using exponent identities

Question: Solve $3^t = 1$

Solution:

Use the identity $a^0 = 1$ : Since any non-zero number raised to the power of 0 equals 1, we can write: $3^t = 3^0$
Apply the same base rule: $t = 0$

Method 2: Taking out common factors

When an exponential equation contains multiple terms with the same base, you can often factor out the common exponential term to simplify the equation.

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Worked Example 3: Factoring method

Question: Solve $5^t + 3 \times 5^{t+1} = 400$

Solution:

Rewrite the expression: Express $5^{t+1}$ as $5^t \times 5$ : $5^t + 3(5^t \times 5) = 400$
Take out the common factor $5^t$ : $5^t(1 + 3 \times 5) = 400$ $5^t(1 + 15) = 400$
Simplify: $5^t(16) = 400$ $5^t = 25$
Express using the same base: Since $25 = 5^2$ : $5^t = 5^2$
Apply the same base rule: $t = 2$

Method 3: Factoring trinomials through substitution

Some exponential equations can be converted into quadratic equations through clever substitution, then solved using factorization techniques.

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Worked Example 4: Trinomial factorization

Question: Solve $3^{2x} - 80 \times 3^x - 81 = 0$

Solution:

Factor the trinomial: This equation can be factored as: $(3^x - 81)(3^x + 1) = 0$
Solve each factor:
- $3^x - 81 = 0$ gives $3^x = 81 = 3^4$ , so $x = 4$
- $3^x + 1 = 0$ gives $3^x = -1$ , which has no real solution since $3^x > 0$ for all real $x$
State the solution: $x = 4$

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Worked Example 5: Substitution with fractional exponents

Question: Solve $p - 13p^{\frac{1}{2}} + 36 = 0$

Solution:

Make a substitution: Notice that $(p^{\frac{1}{2}})^2 = p$ , so let $u = p^{\frac{1}{2}}$ : $u^2 - 13u + 36 = 0$
Factor the quadratic: $(u - 9)(u - 4) = 0$
Solve for $u$ :
- $u - 9 = 0$ gives $u = 9$
- $u - 4 = 0$ gives $u = 4$
Substitute back to find $p$ :
- If $p^{\frac{1}{2}} = 9$ , then $p = 9^2 = 81$
- If $p^{\frac{1}{2}} = 4$ , then $p = 4^2 = 16$
State the solutions: $p = 81 \text{ or } p = 16$

Method 4: Advanced factorization techniques

More complex exponential equations may require sophisticated algebraic manipulation before standard solution methods can be applied.

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Worked Example 6: Using difference of squares

Question: Solve $2^x - 2^{4-x} = 0$

Solution:

Rearrange the equation: $2^x - 2^{4-x} = 0$ $2^x - 2^4 \times 2^{-x} = 0$ $2^x - \frac{2^4}{2^x} = 0$
Eliminate the fraction: Multiply both sides by $2^x$ : $2^{2x} - 16 = 0$ $2^{2x} = 16$
Factor using difference of squares: $(2^x)^2 = 16$ $(2^x - 4)(2^x + 4) = 0$
Solve each factor:
- $2^x - 4 = 0$ gives $2^x = 4 = 2^2$ , so $x = 2$
- $2^x + 4 = 0$ gives $2^x = -4$ , which has no real solution since $2^x > 0$
State the solution: $x = 2$

Exam tips

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Exam Tips

Always check if you can express both sides using the same base - this is often the quickest method
Look for opportunities to factor out common exponential terms before attempting other methods
When you see expressions like $a^{2x}$ and $a^x$ together, consider substitution to create a quadratic equation
Remember that $a^x > 0$ for all real $x$ when $a > 0$ , so solutions giving negative values for exponential expressions should be rejected
Be careful with the condition $a \neq 1$ in the same base rule - if the base is 1, the equation may have infinitely many solutions or no solution

Remember!

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Key Points to Remember:

Exponential equations have the unknown variable in the exponent, not the base
Same base rule: If $a^x = a^y$ where $a > 0$ and $a \neq 1$ , then $x = y$
Factor out common exponential terms when multiple terms share the same base
Use substitution to convert complex exponential equations into familiar quadratic forms
Check your solutions by substituting back into the original equation

Exponential Equations (Grade 10 NSC Matric Mathematics): Revision Notes