Problem 121 - Algebra and Trigonometry

Question

\( \text { Solve: } \quad 8^{x+5}=4^{x-1}\)

Step-by-Step Solution

Answer

The solution to the equation \(8^{x+5} = 4^{x-1}\) is \(x = -17\)

Step 1: Rewrite Bases to a Common Base

The equation \(8^{x+5} = 4^{x-1}\) can be rewritten because both 8 and 4 are powers of 2: \(2^{3(x+5)} = 2^{2(x-1)}\). So this translates to \(2^{3x+15} = 2^{2x-2}\).

Step 2: Equate Exponents

Since the bases are equal, we can make the exponents equal to each other \(3x+15 = 2x-2\)

Step 3: Solve for x

Subtract \(2x\) from both sides one will have \(x + 15 = -2\). Next, subtract 15 from both sides to solve for \(x\), one will get \(x = -17\)

Key Concepts

Exponents

Exponents are a way of expressing repeated multiplication of the same number, known as the base. For instance, in the expression \(8^{x+5}\), 8 is the base, and \(x+5\) is the exponent. The meaning behind this expression is that 8 is multiplied by itself \((x+5)\) times.

• The notation \(a^n\) signifies the base \(a\) is being multiplied by itself \(n-1\) more times.
• Understanding exponents is crucial as they offer shortcuts in mathematical calculations and are especially pivotal in algebra.
• Exponents follow specific rules such as multiplication \((a^m \times a^n = a^{m+n})\), division \((a^m \div a^n = a^{m-n})\), and raising a power to another power \((a^{m^n} = a^{m \times n})\).

Solving Equations

In algebra, solving equations means finding the value of the variable that makes the equation true. When presented with equations involving exponents, such as \(8^{x+5} = 4^{x-1}\), additional strategies such as rewriting exponential expressions or finding common bases can be employed.

• The goal is to isolate the variable, in this case \(x\), to find its value.
• Steps include simplification of expressions, using operations such as addition, subtraction, multiplication, and division.
• Equation solving often involves techniques like factoring or using the properties of equal bases.

Common Base

A common base is used when dealing with equations involving exponents that do not initially share the same base. In our exercise, the numbers 8 and 4 are rewritten to have the same base of 2 because both are powers of 2 (\(8 = 2^3\) and \(4 = 2^2\)).

• The equation becomes \(2^{3(x+5)} = 2^{2(x-1)}\).
• This simplification is possible because powers of a number can be manipulated easily once the bases are the same.
• Finding a common base allows the comparison of exponents, leading to a straightforward solution.

Equating Exponents

Equating exponents is an efficient strategy when solving equations with a common base. Once the bases on both sides of the equation are identical, the exponents themselves can be equated. This is because if \(a^m = a^n\), then it must be that \(m = n\) for the equality to hold.

• In the equation \(2^{3x+15} = 2^{2x-2}\), the exponents \(3x+15\) and \(2x-2\) must be equal.
• This leads to the equation \(3x+15 = 2x-2\), which can be solved using basic algebraic operations.
• By carefully equating and simplifying, we arrive at the solution \(x = -17\).