Problem 82 - Calculus for Biology and Medicine

Question

Solve for \(x\). (a) \(5^{x}=625\) (b) \(4^{4 x}=256\) (c) \(10^{2 x}=0.0001\)

Step-by-Step Solution

Answer

(a) x = 4, (b) x = 1, (c) x = -2.

Step 1: Recognize the Form of the Equation

Notice that all equations are of the form \( a^b = c \). We'll need to represent \( c \) as a power of \( a \) in each case to solve for \( x \).

Step 2: Solve Part (a) by Expressing 625 as a Power of 5

Recognize that 625 can be written as \( 5^4 \), so we can rewrite the equation as: \[ 5^x = 5^4 \]By equating the exponents since the bases are the same, we find \( x = 4 \).

Step 3: Solve Part (b) by Expressing 256 as a Power of 4

Recognize that 256 is \( 4^4 \). Rewrite the equation:\[ 4^{4x} = 4^4 \]Equate the exponents:\[ 4x = 4 \]Divide both sides by 4 to solve for \( x \):\[ x = 1 \]

Step 4: Solve Part (c) by Expressing 0.0001 as a Power of 10

Recognize that 0.0001 is \( 10^{-4} \). Rewrite the equation:\[ 10^{2x} = 10^{-4} \]Equate the exponents:\[ 2x = -4 \]Divide both sides by 2 to solve for \( x \):\[ x = -2 \]

Key Concepts

Solving for x

When we talk about solving for \(x\) in an equation like \(a^b = c\), our goal is to find out the value of \(x\) that makes the equation true. To do this, we often need to adjust the equation so that it's easy to see what \(x\) is. This usually involves expressing \(c\) as a power of \(a\). There are a few key steps one might follow:

• Express the number on the right-hand side of the equation as a power of the base on the left-hand side.
• Rewrite the equation so both sides have the same base with the exponents clearly visible.
• From here, equate the exponents (the numbers the base is raised to) and solve for \(x\).

These steps help make even tricky problems more straightforward by breaking them into manageable pieces. By focusing on what \(x\) needs to be so the equation balances, the process becomes a simple application of basic algebra.

Equating Exponents

Equating exponents is a strategy used when you have equations with the same base on both sides. When two powers of the same base are equal, the exponents themselves must also be equal.Imagine you have the equation \(a^x = a^y\). Since both sides have the same base \(a\), the exponents \(x\) and \(y\) must be the same for the equation to hold true. Here's a simple breakdown:

• If you have transformed both sides of your equation to have the same base, the complicated part is done!
• Now, you only need to set the exponents equal to each other (i.e., \(x = y\)).
• This makes solving for the variable very straightforward, as it often results in a simple linear equation.

By focusing on matching the exponents, the process of solving exponential equations becomes much simpler and more efficient.

Powers of Numbers

Understanding powers of numbers is crucial when dealing with exponential equations. The power of a number refers to how many times you multiply that number by itself. For example:

• \(2^3\) means \(2 \times 2 \times 2\), which gives 8.
• \(5^4\) means \(5 \times 5 \times 5 \times 5\), resulting in 625.
• Powers can also be negative, meaning you divide 1 by the base raised to the positive of the power, such as \(10^{-3} = \frac{1}{10^3} = 0.001\).

Working with powers helps in expressing large or small numbers in a condensed form, making computations easier.Knowing how to identify or transform numbers into powers of other numbers is a highly useful skill in mathematics. This allows you to easily manipulate and solve equations like those posed in the original exercise, by setting up both sides of equations with common bases.