Exponential Equations | General Mathematics

Summary

This video by Teacher John on 'General Math Made Easy' teaches how to solve exponential equations. It explains that an exponential equation involves one or both sides containing exponential expressions and highlights the key rule: if the bases are the same, the exponents are equal. The lesson walks through multiple examples, demonstrating how to identify same bases, apply exponent laws, isolate the variable, and verify solutions. Techniques include rewriting expressions with common bases, using negative exponents, and solving quadratic forms when necessary. The video concludes by reinforcing essential rules and encouraging students to practice and verify their solutions for mastery.

Takeaways

• 😀 An exponential equation is one where at least one side contains an exponential expression.
• 😀 The main rule: if a^b = a^c, then b = c, meaning equal bases allow equating exponents.
• 😀 Always check if both sides of the equation have the same base before solving.
• 😀 If the bases are not the same, rewrite them as powers of the same base or use logarithms.
• 😀 Solve for the unknown variable by isolating it after equating exponents.
• 😀 Verify your solution by substituting it back into the original equation to ensure both sides are equal.
• 😀 The product law states: a^n * a^m = a^(n+m), useful when combining exponential terms.
• 😀 The negative exponent law states: 1 / a^n = a^(-n), useful for rewriting fractions as exponents.
• 😀 Exponential equations can sometimes form quadratic equations, requiring factoring to solve.
• 😀 Always simplify and combine constants before equating exponents for easier calculations.
• 😀 Multiple solutions are possible; check all solutions against the original equation.
• 😀 Examples include solving equations like 6^(x+5) = 6^3 and 9^(x^2) = 3^(x+3), demonstrating different solving strategies.

Q & A

What is an exponential equation?

An exponential equation is an equation where one or both sides contain exponential expressions, meaning the variable appears in the exponent.

What is the key rule for solving exponential equations when the bases are the same?

If a^b = a^c, then b = c. This means that when the bases are the same, you can equate the exponents to solve for the unknown variable.

What should you do if the bases on both sides of an exponential equation are not the same?

You should try to rewrite one or both sides as exponential expressions with the same base. If this is not possible, logarithms must be used to solve the equation.

What is the first step in solving an exponential equation?

The first step is to determine whether both sides of the equation have the same base, in order to equate the exponents.

How do you solve the equation 6^(x+5) = 6^3?

Since the bases are the same, equate the exponents: x + 5 = 3. Solve for x to get x = -2. Check by substituting back into the original equation.

What exponent rules are commonly used when solving exponential equations?

The main exponent rules used are: the Product Rule (a^m * a^n = a^(m+n)) and the Negative Exponent Rule (1/a^n = a^-n).

How do you solve 16^-x = 1/64 using exponent rules?

Rewrite 1/64 as 64^-1, then express both 16 and 64 as powers of 4: 16^-x = 4^-2x, 64^-1 = 4^-3. Equate exponents: -2x = -3 → x = 3/2.

What should you do after finding the solution of an exponential equation?

Always check the solution by substituting the value back into the original equation to ensure both sides are equal.

How do you handle an exponential equation that results in a quadratic equation?

Rewrite the equation with the same base, equate exponents, and if the exponent expression is quadratic, rearrange it to standard quadratic form and solve by factoring or other quadratic methods.

What is the solution to the equation 9^(x^2) = 3^(x+3)?

Rewrite 9 as 3^2: 3^(2x^2) = 3^(x+3). Equate exponents: 2x^2 = x + 3 → 2x^2 - x - 3 = 0. Factor: (2x-3)(x+1)=0. Solutions: x = 3/2 or x = -1.

Why is it important to rewrite fractional bases using negative exponents?

Using negative exponents allows us to express fractions as standard exponential expressions with positive bases, making it easier to equate exponents and solve the equation.

What are the three main steps in solving any exponential equation?

1. Make bases the same if possible. 2. Equate exponents and solve for the variable. 3. Check the solution by substituting back into the original equation.