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 $$2.113$$.