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Chapter 5: Problem 121

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

Short Answer

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

Step by step solution

01

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 \(2^{3x+15} = 2^{2x-2}\).
02

Equate Exponents

Since the bases are equal, we can make the exponents equal to each other \(3x+15 = 2x-2\)
03

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 \(x = -17\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 \(x = -17\).

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Most popular questions from this chapter

In Exercises \(113-116\), use the keys on your calculator or graphing utility for converting an angle in degrees, minutes, and seconds \(\left(D^{\circ} M^{\prime} S^{\prime \prime}\right)\) into decimal form, and vice versa. In Exercises \(113-114\), convert each angle to a decimal in degrees. Round your answer to two decimal places. $$ 65^{\circ} 45^{\prime} 20^{\prime \prime} $$

Use a right triangle to write each expression as an algebraic expression. Assume that \(x\) is positive and that the given inverse trigonometric function is defined for the expression in \(x\). $$ \sec \left(\cos ^{-1} \frac{1}{x}\right) $$

Describe the restriction on the tangent function so that it has an inverse function.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because \(y=\sin x\) has an inverse function if \(x\) is restricted to \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right],\) they should make restrictions easier to remember by also using \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) as the restriction for \(y=\tan x\).

Describe the restriction on the cosine function so that it has an inverse function.
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