Exponential Equations Revision Notes for Grade 12 NSC Matric Mathematics

Exponential Equations

What are exponential equations?

Exponential equations are mathematical equations where the unknown variable appears in the exponent position. These equations require special solving techniques because the variable is "hidden" in the power rather than being a simple coefficient or base.

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The key to solving exponential equations lies in expressing all terms using the same base, then equating the exponents to find the solution.

Laws of exponents (recap)

Before tackling exponential equations, you need to master these fundamental laws of exponents:

Product rule: $a^m \times a^n = a^{m+n}$
Quotient rule: $a^m \div a^n = a^{m-n}$
Power rule: $(a^m)^n = a^{m \times n}$
Negative exponent rule: $a^{-m} = \frac{1}{a^m}$
Zero exponent rule: $a^0 = 1$

These rules allow you to manipulate exponential expressions and prepare them for solving.

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Master these five laws completely - they are your essential tools for solving any exponential equation!

Case 1: Same base on both sides

When both sides of an exponential equation have the same base, you can directly equate the exponents and solve.

Method: If both sides have identical bases, equate the exponents and solve for the unknown variable.

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Worked Example: Same Base Method

Solve for x: $2^x = 8$

Step 1: Express 8 as a power of 2 $8 = 2^3$

Step 2: Rewrite the equation $2^x = 2^3$

Step 3: Since the bases are identical, equate the exponents $x = 3$

Case 2: Different bases - convert to prime bases

When the equation has different bases, convert all terms to their prime factor bases before solving.

Method: Convert composite numbers to their prime factorisation, then apply the laws of exponents.

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Worked Example: Prime Factor Conversion

Solve for x: $5^{x+1} = 125^{x-3}$

Step 1: Express 125 as a power of 5 $125 = 5^3$

Step 2: Rewrite the equation $5^{x+1} = (5^3)^{x-3}$

Step 3: Apply the power rule $5^{x+1} = 5^{3(x-3)}$ $5^{x+1} = 5^{3x-9}$

Step 4: Equate the exponents $x + 1 = 3x - 9$

Step 5: Solve for x $x + 1 = 3x - 9$ $10 = 2x$ $x = 5$

Case 3: Negative exponents

When dealing with negative exponents, convert them to fractional forms using the negative exponent rule.

Method: Use fractional forms when necessary, applying $a^{-m} = \frac{1}{a^m}$ .

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Worked Example: Negative Exponents

Solve for x: $2^x = \frac{1}{8}$

Step 1: Express 8 as a power of 2 $8 = 2^3$ , so $\frac{1}{8} = 2^{-3}$

Step 2: Rewrite the equation $2^x = 2^{-3}$

Step 3: Equate exponents $x = -3$

Case 4: Quadratic form in exponents

When equations contain terms like $a^{2x}$ and $a^x$ , use substitution to create a quadratic equation.

Method: If the equation contains terms like $a^{2x}$ and $a^x$ , set $a^x$ as a substitution variable.

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This method is particularly useful when you see exponential expressions that can be factored or simplified algebraically.

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Worked Example: Quadratic Form

Solve for x: $5^x + 5^{x+1} = 30$

Step 1: Factor out $5^x$ $5^x(1 + 5) = 30$ $5^x(6) = 30$

Step 2: Solve for $5^x$ $5^x = \frac{30}{6} = 5$

Step 3: Since $5^x = 5^1$ , equate exponents $x = 1$

Additional worked examples

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Worked Example 1: Mixed Base Conversion

Solve for x: $3^{9-2x} = 27^{x-1}$

Solution:

Step 1: Express 27 as a power of 3 $27 = 3^3$

Step 2: Rewrite the equation $3^{9-2x} = (3^3)^{x-1}$

Step 3: Apply the power rule $3^{9-2x} = 3^{3(x-1)}$ $3^{9-2x} = 3^{3x-3}$

Step 4: Equate the exponents $9 - 2x = 3x - 3$ $12 = 5x$ $x = \frac{12}{5}$

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Worked Example 2: Power of 10

Solve for x: $10^{x+1} = 100$

Solution:

Step 1: Express 100 as a power of 10 $100 = 10^2$

Step 2: Rewrite the equation $10^{x+1} = 10^2$

Step 3: Equate the exponents $x + 1 = 2$ $x = 1$

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Always check your answers by substituting back into the original equation to verify your solution is correct!

Summary of key steps

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Key Steps for Solving Exponential Equations:

Convert all terms to prime factor bases
Use laws of exponents to simplify expressions
Equate exponents when bases are identical
Use substitution for quadratic-like expressions
Apply factorisation techniques where necessary

Remember!

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Essential Points to Remember:

Exponential equations have variables in the exponent - identify them first
Always try to express both sides using the same base before solving
Master the five laws of exponents - they are your key tools
For complex forms, consider substitution or factorisation methods
Check your answers by substituting back into the original equation

Exponential Equations (Grade 12 NSC Matric Mathematics): Revision Notes