In this mini-lesson, we will explore the world of exponential equations by understanding the rules for solving exponents with same base, using logarithm, and we will learn how to apply them while solving problems. We will also discover interesting facts around them.

We know that the exponent of a number indicates the number of times the number (base) is multiplied by itself.

But, what happens if the power of a number is variable?

Exponential equations play an important role in solving such problems.

The above example is a type of exponential equation.

Now, let's explore exponential equations in more detail.

**Lesson Plan**

**What Are Exponential Equations? **

Equations like \(3^{x} = 9\), \((-4)^{2x + 1} = 64\), and \(x^y = 36\) are examples of exponential equations.

**Definition**

Exponential equations are the type of equations which contain a base and variables as the exponent.

The equation has two sides and we need to solve for the variables using the conditions given in the question.

**What Is the Formula for Exponential Equations?**

There are different types of exponential equations in mathematics.

Some have equal bases on both sides of the equation and some involve different bases.

We also need to use logarithms to solve the equations.

So, there is no particular formula to solve exponential equations; each question comes with a different case.

We will be solving several types of exponential equations in this lesson.

**How to Solve Exponential Equations?**

Solving an exponential equation is as easy as performing any other mathematical calculations.

An exponential equation is solved by making the base same on both sides of the equation.

An exponential equation can also be solved by using logarithms.

**Solving Exponential Equations With Logarithms**

Before we begin solving an exponential equation with logarithm, let's first explore some logarithmic identities.

**Identity - 1**

If \[\log_a b = c\]

then \[a^c = b\]

**Identity - 2**

\[\log_a a^n = n\]

**Identity - 3**

\[\log_a (m \times n) = \log_a m + \log_a n\]

Let's solve exponential equations examples.

**Example - 1**

Solve for \(x\) when \(10^x = 200\)

**Solution**

\[\begin{align}10^x &= 200 \\

\text{Taking log } &\text{both sides} \\

\log_{10} 10^x &= \log_{10} 200 \\

x \times \log_{10} 10 &= \log_{10} (2 \times 100) \\

x \times 1 &= \log_{10} 2 + \log_{10} 100 \\

x &= \log_{10} 2 + \log_{10} 10^2 \\

x &= \log_{10} 2 + 2 \\

x &= 2 + \log_{10} 2 \end{align}\]

**Example - 2**

Find the value of \(x\) in the given equation.

\(2^{2x - 1} = 6^x\)

**Solution**

\[\begin{align}2^{2x - 1} &= 6^x \\

\text{Taking log on} &\text{both sides} \\[0.2cm]

\log_{2} 2^{2x - 1} &= \log_{2} 6^x \\[0.2cm]

(2x - 1) \times \log_{2} 2 &= \log_{2} (2^x \times 3^x) \\[0.2cm]

(2x - 1) \times 1 &= \log_{2} 2^x + \log_{2} 3^x \\[0.2cm]

2x - 1 &= x + x\log_{2} 3 \\[0.2cm]

2x - x - 1 &= x\log_{2} 3 \\[0.2cm]

x - 1 &= x\log_{2} 3 \\[0.2cm]

x - x\log_{2} 3 &= 1 \\[0.2cm]

x(1 - \log_{2} 3) &= 1 \\[0.2cm]

x &= \frac{1}{1 - \log_{2} 3}\end{align}\]

**Solving Exponential Equations With Same Base**

Before we solve an exponential equation with the same base, we need to remember that if the bases are equal, then the exponents must be equal.

Let's solve exponential equations examples.

**Example - 1**

Solve for \(x\), \(5^x = 5^4\)

**Solution**

We know that if the base is the same, the powers must be equal.

Thus, \(x = 4\)

**Example - 2**

Find the value of \(x\) in the given equation.

\(4^{2x - 1} = 4^{1 - x}\)

**Solution**

We know that if the base is the same, the powers must be equal.

\[\begin{align}4^{2x - 1} &= 4^{1 - x} \\

2x - 1 &= 1 - x \\[0.2cm]

2x + x &= 1 + 1 \\[0.2cm]

3x &= 2 \\[0.2cm]

x &= \frac{2}{3}\end{align}\]

- If for any equation \(a^m = a^n\), where \(a \neq 0\), then \(m = n\).
- The log for any number of the form \(a^m\) to the base \(a\) is \(\log_a a^m = m\).
- If zero is the exponent of any number, its value is 1 i.e., \(a^0 = 1\).

**Solving Exponential Equations With Unlike Bases**

When we solve exponential equations with different bases, we usually come across the following two cases.

**When Base Can be Made Same**

Always check while solving exponential equations with different bases whether the bases can be made the same or not.

If the bases can be made the same, follow the steps shown in the example below.

**Example**

Solve for \(y\) in \(7^{y+1} = 343^y\)

**Solution**

\[\begin{align}7^{y+1} &= 343^y \\[0.2cm]

7^{y+1} &= {(7^3)}^{y} \\[0.2cm]

7^{y+1} &= 7^{3y} \\[0.2cm]

\text{Base are the same, } & \text{let us equate the powers}\\[0.2cm]

y + 1 &= 3y \\[0.2cm]

1 &= 3y - y \\[0.2cm]

1 &= 2y \\[0.2cm]

y &= \frac{1}{2}\end{align}\]

**When Base Cannot be Made Same**

If the bases to both sides of the equation cannot be made same, we have to apply the logarithmic methods to solve the equation.

**Example**

Solve for \(a\) in \(5^{3a-1} = (\dfrac{1}{500})^{1-a}\)

**Solution**

\(5^{3a-1} = (\dfrac{1}{500})^{1-a} \\[0.2cm]\)

\(5^{3a-1} = (500^{-1})^{1-a} \\[0.2cm]\)

\(5^{3a-1} = 500^{-1 \times 1-a} \\[0.2cm]\)

\(5^{3a-1} = 500^{a - 1} \\[0.2cm]\)

\(\text{Base cannot be made the same.}\)

\(\text{Let us take log with base 5 on both sides}\\[0.2cm]\)

\(\log_5 5^{3a-1} = \log_5 500^{a - 1}\\[0.2cm]\)

\((3a - 1)\log_5 5 = \log_5 ({(125 \times 4)}^{a - 1})\\[0.2cm]\)

\((3a - 1)\log_5 5 = \log_5 (125^{a - 1} \times 4^{a - 1})\\[0.2cm]\)

\((3a - 1)\log_5 5 = \log_5 (125^{a - 1}) + \log_5 4^{a - 1}\\[0.2cm]\)

\((3a - 1)\log_5 5 = \log_5 5^{3(a - 1)} + \log_5 2^{2(a - 1)}\\[0.2cm]\)

\((3a - 1) \times 1 = 3(a-1) + 2(a-1)\log_5 2\\[0.2cm]\)

\(3a - 1 = 3a - 3 + 2(a-1)\log_5 2\\[0.2cm]\)

\(3 - 1 = 2(a-1)\log_5 2\\[0.2cm]\)

\(2 = 2(a-1)\log_5 2\\[0.2cm]\)

\(1 = (a-1)\log_5 2\\[0.2cm]\)

\(\frac{1}{\log_5 2} = a-1\\[0.2cm]\)

\(\log_2 5 = a-1\\[0.2cm]\)

\(\text{Using the } \text{identity}\\[0.2cm]\)

\(\frac{1}{log_a b} = \log_b a \\[0.2cm]\)

\(a = \log_2 5 + 1\)

- Solve for \(x\) in the equation \(4^x + 6^x = 9^x\)

**Solved Examples**

Example 1 |

Help Tim find the value of \(x\) for the given exponential equation \(7^{3x+7} = 490\)

**Solution**

\( 7^{3x+7} = 490 \\[0.2cm]\)

\(\text{Base cannot be made same; } \)

\( \text{let us take log with base 7 on both sides}\\[0.2cm]\)

\(\log_7 (7^{3x+7}) = \log_7 490 \\[0.2cm]\)

\((3x+7)\log_7 7 = \log_7 (49 \times 10) \\[0.2cm]\)

\((3x+7)\log_7 7 = \log_7 49 + \log_7 10 \\[0.2cm]\)

\((3x+7) \times 1 = \log_7 7^2 + \log_7 10 \\[0.2cm]\)

\(3x+7 = 2\log_7 7 + \log_7 10 \\[0.2cm]\)

\(3x+7 = 2 \times 1 + \log_7 10 \\[0.2cm]\)

\(3x+7 = 2 + \log_7 10 \\[0.2cm]\)

\(3x = 2 - 7 + \log_7 10 \\[0.2cm]\)

\(3x = -5 + \log_7 10 \\[0.2cm]\)

\(x = \frac{\log_7 10 - 5}{3}\)

\(\therefore\) Answer is \(x = \dfrac{\log_7 10 - 5}{3}\) |

Example 2 |

Help Clara find the value of \(x\) in \(\dfrac{27}{{3^{-x}}} = 3^6\)

**Solution**

Here we have negative exponents with variables.

\[\begin{align}\frac{27}{{3^{-x}}} &= 3^6 \\[0.2cm]

\frac{3^3}{{3^{-x}}} &= 3^6 \\[0.2cm]

3^3 \times 3^x &= 3^6 \\[0.2cm]

3^{(3 +x)} &= 3^6\end{align}\]

If bases are the same, then exponents must be equal.

\(x + 3 = 6\)

\(x = 3\)

\(\therefore\) Value of \(x\) is \(3\) |

Example 3 |

Find the value of \(x\) in the exponential equation: \(121^{2x-1} = (\sqrt{11})^{3x+4}\)

**Solution**

\[\begin{align} 121^{2x-1} &= (\sqrt{11})^{3x+4} \\[0.2cm]

(11^2)^{2x-1} &= (11^{\frac{1}{2}})^{3x+4} \\[0.2cm]

(11)^{2(2x-1)} &= 11^{\frac{3x+4}{2}} \\[0.2cm]

\text{Base are same, } & \text{then powers are equal}\\[0.2cm]

2(2x - 1) &= \frac{3x+4}{2} \\[0.2cm]

4x - 2 &= \frac{3x+4}{2} \\[0.2cm]

(4x - 2) \times 2 &= 3x+4 \\[0.2cm]

8x - 4 &= 3x+4 \\[0.2cm]

8x - 3x &= 4 + 4 \\[0.2cm]

5x &= 8 \\[0.2cm]

x &= \frac{8}{5}\end{align}\]

\(\therefore\) Answer is \(x = \dfrac{8}{5}\) |

**Interactive Questions**

**Here are a few activities for you to practice. **

**Select/Type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize**

The mini-lesson targeted in the fascinating concept of exponential equations. The math journey around exponential equations starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

**About Cuemath**

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Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we at Cuemath believe in.

**Frequently Asked Questions (FAQs)**

## 1. What are negative exponents?

Negative exponents are nothing but the numbers which have negative powers or negative exponents.

For example: \(5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}\)

So, \(5^{-2}\) is a negative exponent.