Solve The Equation:$\[ 10^{-3x} \cdot 10^x = \frac{1}{10} \\]

Feb 28, 2025 by ADMIN 62 views

**Solving Exponential Equations: A Step-by-Step Guide**

Introduction

Exponential equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving the equation $10^{-3x} \cdot 10^x = \frac{1}{10}$ , which involves exponential terms with different bases and exponents. We will break down the solution into manageable steps, making it easy to understand and follow along.

Understanding Exponential Terms

Before we dive into solving the equation, let's take a closer look at exponential terms. An exponential term is a mathematical expression that represents a quantity that grows or decays at a constant rate. In the equation $10^{-3x} \cdot 10^x = \frac{1}{10}$ , we have two exponential terms: $10^{-3x}$ and $10^x$ . The base of these exponential terms is 10, and the exponents are $-3x$ and $x$ , respectively.

Simplifying the Equation

To solve the equation, we need to simplify it first. We can start by using the properties of exponents to combine the two exponential terms on the left-hand side of the equation. Specifically, we can use the rule that states $a^m \cdot a^n = a^{m+n}$ .

import math

# Define the equation
def equation(x):
    return 10**(-3*x) * 10**x

# Simplify the equation using the properties of exponents
def simplified_equation(x):
    return 10**(-2*x)

By applying the rule, we can simplify the equation to $10^{-2x} = \frac{1}{10}$ .

Isolating the Variable

Now that we have simplified the equation, we can isolate the variable $x$ . To do this, we need to get rid of the exponent $-2x$ on the left-hand side of the equation. We can do this by taking the logarithm of both sides of the equation.

import math

# Define the equation
def equation(x):
    return 10**(-2*x) - 1/10

# Take the logarithm of both sides of the equation
def isolated_variable():
    return math.log(1/10, 10) / -2

By taking the logarithm of both sides of the equation, we can isolate the variable $x$ and solve for its value.

Solving for x

Now that we have isolated the variable $x$ , we can solve for its value. We can do this by evaluating the expression on the right-hand side of the equation.

import math

# Define the equation
def equation(x):
    return math.log(1/10, 10) / -2

# Solve for x
def solve_for_x():
    return equation(0)

By evaluating the expression on the right-hand side of the equation, we can find the value of $x$ that satisfies the equation.

Conclusion

In this article, we have solved the equation $10^{-3x} \cdot 10^x = \frac{1}{10}$ using the properties of exponents and logarithms. We have broken down the solution into manageable steps, making it easy to understand and follow along. By applying the rules of exponents and logarithms, we have isolated the variable $x$ and solved for its value. This solution can be applied to a wide range of problems involving exponential equations.

Additional Resources

For more information on solving exponential equations, check out the following resources:

Frequently Asked Questions

Q: What is the difference between an exponential equation and a logarithmic equation? A: An exponential equation involves a variable in the exponent, while a logarithmic equation involves a variable as the exponent.
Q: How do I simplify an exponential equation? A: You can simplify an exponential equation by using the properties of exponents, such as the rule that states $a^m \cdot a^n = a^{m+n}$ .
Q: How do I isolate a variable in an exponential equation? A: You can isolate a variable in an exponential equation by taking the logarithm of both sides of the equation.

Glossary

Exponential term: A mathematical expression that represents a quantity that grows or decays at a constant rate.
Logarithmic term: A mathematical expression that represents the power to which a base number must be raised to produce a given value.
Property of exponents: A rule that describes how to combine or simplify exponential terms.
Isolate a variable: To get a variable by itself on one side of an equation, without any other variables or constants attached to it.
Frequently Asked Questions: Exponential Equations =====================================================

Q: What is an exponential equation?

A: An exponential equation is a mathematical equation that involves an exponential term, which is a variable or expression raised to a power. Exponential equations can be written in the form $a^x = b$ , where $a$ is the base, $x$ is the exponent, and $b$ is the result.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you can use the properties of exponents, such as the rule that states $a^m \cdot a^n = a^{m+n}$ . You can also use the rule that states $(a^m)^n = a^{m \cdot n}$ .

Q: How do I isolate a variable in an exponential equation?

A: To isolate a variable in an exponential equation, you can take the logarithm of both sides of the equation. This will allow you to get rid of the exponent and solve for the variable.

Q: What is the difference between an exponential equation and a logarithmic equation?

A: An exponential equation involves a variable in the exponent, while a logarithmic equation involves a variable as the exponent. For example, the equation $2^x = 8$ is an exponential equation, while the equation $x = \log_2 8$ is a logarithmic equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use the properties of exponents and logarithms. You can also use the rule that states $a^x = b$ can be rewritten as $x = \log_a b$ .

Q: What is the rule for multiplying exponential terms?

A: The rule for multiplying exponential terms is $a^m \cdot a^n = a^{m+n}$ .

Q: What is the rule for raising an exponential term to a power?

A: The rule for raising an exponential term to a power is $(a^m)^n = a^{m \cdot n}$ .

Q: How do I evaluate an exponential expression?

A: To evaluate an exponential expression, you can use the rule that states $a^m = a \cdot a \cdot a \cdot ... \cdot a$ (m times).

Q: What is the difference between an exponential function and an exponential equation?

A: An exponential function is a mathematical function that involves an exponential term, while an exponential equation is a mathematical equation that involves an exponential term. For example, the function $f(x) = 2^x$ is an exponential function, while the equation $2^x = 8$ is an exponential equation.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a computer program. You can also use the rule that states the graph of an exponential function is a curve that approaches the x-axis as x approaches negative infinity.

Q: What is the rule for dividing exponential terms?

A: The rule for dividing exponential terms is $\frac{a^m}{a^n} = a^{m-n}$ .

Q: What is the rule for subtracting exponential terms?

A: The rule for subtracting exponential terms is $a^m - a^n = a^n (a^{m-n} - 1)$ .

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the rule that states $\log_a b = c$ can be rewritten as $a^c = b$ .

Q: What is the rule for multiplying logarithmic terms?

A: The rule for multiplying logarithmic terms is $\log_a b \cdot \log_a c = \log_a (b \cdot c)$ .

Q: What is the rule for raising a logarithmic term to a power?

A: The rule for raising a logarithmic term to a power is $(\log_a b)^n = \log_a (b^n)$ .

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you can use the rule that states $\log_a b = c$ can be rewritten as $a^c = b$ .

Q: What is the difference between a logarithmic function and a logarithmic equation?

A: A logarithmic function is a mathematical function that involves a logarithmic term, while a logarithmic equation is a mathematical equation that involves a logarithmic term. For example, the function $f(x) = \log_2 x$ is a logarithmic function, while the equation $\log_2 x = 3$ is a logarithmic equation.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use a graphing calculator or a computer program. You can also use the rule that states the graph of a logarithmic function is a curve that approaches the x-axis as x approaches negative infinity.

Q: What is the rule for dividing logarithmic terms?

A: The rule for dividing logarithmic terms is $\frac{\log_a b}{\log_a c} = \log_a \frac{b}{c}$ .

Q: What is the rule for subtracting logarithmic terms?

A: The rule for subtracting logarithmic terms is $\log_a b - \log_a c = \log_a \frac{b}{c}$ .

Q: How do I solve a system of exponential and logarithmic equations?

A: To solve a system of exponential and logarithmic equations, you can use the rules for exponential and logarithmic equations, as well as the rules for solving systems of equations.