# Exponential equations

**What are exponential equations?**

The exponential function is that function that is represented in the form $$x^{m}$$ where $$x$$ is base and $$m$$ is the exponent.

The exponential curve depends on the exponential function, and it also depends on the value of $$m$$ in the expression $$x^{m}$$.

A real-life example is fire. Fire shows the exponential growth in the forests.

**E1.7A: Understand the meaning of indices (fractional, negative and zero) and use the rules of the indices. **

There are eight laws of indices which is going to be discussed below.

**Law 1:** If one variable suppose $$x$$ is there with a different power as $$x^{m}$$ and $$x^{n}$$, then multiplication of both the variables will be $$x^{(m+n)}$$.

**Law 2:** If one variable with different power as $$x^{m}$$ and $$x^{n}$$, then the division of both the variables will be $$x^{(m-n)}$$.

**Law 3:** If one variable contains the power of power like $$({x^{m})}^{n}$$ which will be $$x^{m\cdot n}$$.

**Law 4:** If one variable suppose $$x$$ which contains zero as $$x^{0}$$ then it is equal to $$1$$.

**Law 5:** If one variable with negative power $$x^{-m}$$ which is equal to $$\frac{1}{x^{m}}$$.

**Law 6:** If one variable which is having a power in fraction like $$x^{\frac{m}{n}}$$, then it is equal to $$(\sqrt[n]{x})^{m}$$.

**Law 7:** If there are two variables, suppose $$x$$ and $$y$$ with the same power $$m$$, then the product of both the variables will be $$(x\times y)^{m}$$.

**Law 8:** If there are two variables with the same power, then the division of both the variables will be $$(x\div y)^{m}$$.

**Worked examples**

**Example 1:** Bacteria initially contains only one bacterium and then doubles every hour. Find how many bacteria are in the culture at the end on $$18$$ hours?

**Step 1: Given Information.**

Initially, at $$t=0$$, the number of bacteria is initially equal to $$1$$.

**Step 2: Find the number of bacteria every hour.**

When $$t=0$$, number of bacteria is $$2^{0}$$.

When $$t=1$$, number of bacteria is $$2^{1}$$.

When $$t=0$$, number of bacteria is $$2^{2}$$.

When $$t=0$$, number of bacteria is $$2^{3}$$ and so on.

**Step 3: Write the exponential equation.**

$$f(t)=2^{t}$$.

**Step 4: Calculate the number of bacteria for $$t=18$$.**

Number of bacteria at $$t=18$$ is $$2^{18}=262144$$

Hence the number of bacteria at the end of $$18$$ hours is $$262144$$.

**E1.17: Use the exponential growth and decay in relation to the population and finance.**

**Exponential growth**

Exponential growth is any quantity that initially increases slowly and then rapidly. In this, the rate of change increases over time. It is represented as $$x=m(1+r)^{t}$$ where $$r$$ is the growth percentage.

**Exponential decay**

Exponential decay is a quantity that is initially decreasing rapidly and then slowly. It is represented as $$x=m(1-r)^{t}$$ where $$r$$ is the delay percentage. The population is increasing exponentially, so it shows an exponential growth. Compound interest shows an exponential decay.

**Worked examples**

__ Example 1:__ Rohan deposits $$60000$$ in a bank that pays $$10%$$ compound interest annually. How much money will he have after $$20$$ years without withdrawal?

**Step 1: Use the exponential delay formula as $$x=m(1-r)^{t}$$.**

Initially, at $$t=0$$ investment of $$60000$$ is done.

Then at $$t=1$$ investment of $$60000(1+0.1)^{1}=66000$$ is done.

Then at $$t=2$$ investment of $$60000(1+0.1)^{2}=72600$$ is done.

$$\vdots$$

Then at $$t=20$$ investment of $$60000(1+0.1)^{20}=403649.997$$ is done.

**Step 2: Write an exponential expression.**

$$x=60000(1.1)^{t}$$.

So, he will be having $$403649.997$$ amount of money after $$20$$ years.

**Example 2:** Solve the exponential equation $$3^{4}=2^{(3x)}$$.

**Step 1: Take the natural logarithm on both the sides.**

$$ln{3^{4}}=ln{2^{(3x)}}$$.

It can be written as $$4ln{3}={3x}ln{2}$$.

**Step 2: Solve the above equation for the value of $$x$$.**

$$\frac{4ln{3}}{3ln{2}}=x$$

**Step 3: Simplify $$\frac{4ln3}{3ln2}=x$$ for the value of $$x$$.**

$$x=2.113$$.

Hence the value of $$x$$ after solving $$\frac{4ln{3}}{3ln{2}}=x$$ is $$2.113$$.