## 17Calculus Precalculus - Solve Exponentials

##### 17Calculus

Solving Equations Involving Exponentials

When we have a variable inside a logarithm, we need to move it out of the logarithm to determine it's value. To do that, we use the laws listed on the laws of logarithms page. Here is how it works.

When we have an exponential, say $$e^x$$, to get the variable out of the exponent we take the natural log to get $$\ln (e^x)$$. Then we use the law $$\ln(x^y) = y\ln x$$. This gives us $$\ln (e^x) = x\ln e$$. But $$\ln e = 1$$, so we end up with $$\ln (e^x) = x\ln e = x$$. Let's go through an example to see this in action.

Example

Solve $$2^x = 30$$.

 Solution $$2^x = 30$$ Take the natural log of both sides. $$\ln(2^x) = \ln(30)$$ Use the law $$\ln(x^y) = y\ln x$$ to simplify the left side. $$x\ln(2) = \ln(30)$$ Solve for $$x$$. $$x = \ln(30)/\ln(2)$$

Practice

Unless otherwise instructed, solve these problems using the natural logarithm giving your answers in exact terms.

$$3^x = 5$$

Problem Statement

Solve $$3^x = 5$$

Solution

### 3035 video solution

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$$5^{2x+3} = 8$$

Problem Statement

Solve $$5^{2x+3} = 8$$

Solution

### 3036 video solution

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$$3^{x+2} = 4^{2-x}$$

Problem Statement

Solve $$3^{x+2} = 4^{2-x}$$

Solution

### 3037 video solution

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$$8^x=15$$

Problem Statement

Solve $$8^x=15$$ using the natural logarithm giving your answer in exact terms.

Solution

### PatrickJMT - 1698 video solution

video by PatrickJMT

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$$7^x-1=4$$

Problem Statement

Solve $$7^x-1=4$$ using the natural logarithm giving your answer in exact terms.

Solution

### MIP4U - 1701 video solution

video by MIP4U

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$$3(2^x)-2=13$$

Problem Statement

Solve $$3(2^x)-2=13$$ using the natural logarithm giving your answer in exact terms.

Solution

### MIP4U - 1702 video solution

video by MIP4U

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$$(2/3)^x=5^{3-x}$$

Problem Statement

Solve $$(2/3)^x=5^{3-x}$$ using the natural logarithm giving your answer in exact terms.

Solution

### MIP4U - 1703 video solution

video by MIP4U

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$$5^{x-3}=3^{2x+1}$$

Problem Statement

Solve $$5^{x-3}=3^{2x+1}$$ using the natural logarithm giving your answer in exact terms.

Solution

### MIP4U - 1704 video solution

video by MIP4U

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$$1111=5(2^t)$$

Problem Statement

Solve $$1111=5(2^t)$$ using the natural logarithm giving your answer in exact terms.

Solution

### Khan Academy - 1700 video solution

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$$\displaystyle{\left( \frac{4}{5} \right)^x = 6^{1-x}}$$

Problem Statement

Solve $$\displaystyle{\left( \frac{4}{5} \right)^x = 6^{1-x}}$$ using the natural logarithm giving your answer in exact terms.

Solution

### PatrickJMT - 1699 video solution

video by PatrickJMT

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The exponential function $$V(x) = 25 (4/5)^x$$ is used to model the value of a car over time. Use the properties of logarithms and exponentials to rewrite the model in the form $$V(t) = 25e^{kt}$$.

Problem Statement

The exponential function $$V(x) = 25 (4/5)^x$$ is used to model the value of a car over time. Use the properties of logarithms and exponentials to rewrite the model in the form $$V(t) = 25e^{kt}$$.

$$V(t) = 25e^{t\ln(4/5)}$$

Problem Statement

The exponential function $$V(x) = 25 (4/5)^x$$ is used to model the value of a car over time. Use the properties of logarithms and exponentials to rewrite the model in the form $$V(t) = 25e^{kt}$$.

Solution

Notice that the two equations are the same except for the exponential terms. So we need to determine $$k$$ from $$(4/5)^x = e^{kt}$$. Although not stated in the problem, since the function names are both V, we can assume that $$x=t$$.

 $$(4/5)^x = e^{kt}$$ $$\ln[(4/5)^x] = \ln[e^{kt}]$$ $$x\ln[(4/5)] = kt\ln[e]$$ $$x\ln[(4/5)] = kt$$ $$x\ln[(4/5)] = kt$$ $$(x/t)\ln[(4/5)] = k$$ Since $$x=t$$, $$x/t = 1$$ $$\ln[(4/5)] = k$$

$$V(t) = 25e^{t\ln(4/5)}$$

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