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Chapter 11: Problem 12

Solve each equation. $$\ln (x)+\ln (3)=0$$

Short Answer

Expert verified
x = \frac{1}{3}

Step by step solution

01

- Combine the logarithms

Use the property of logarithms that allows combining sums: \ \ \ \ \[ \ln (a) + \ln (b) = \ln(ab) \] \ \ This gives us: \ \ \[ \ln (x) + \ln (3) = \ln (3x) \] \ \ So the equation becomes: \ \ \[ \ln (3x) = 0 \]
02

- Exponentiate both sides

To eliminate the natural logarithm, exponentiate both sides of the equation using the base of the natural logarithm, which is the constant e: \ \ \[ e^{\ln(3x)} = e^{0} \] \ \ Simplifying, we get: \ \ \[ 3x = 1 \]
03

- Solve for x

Divide both sides by 3 to isolate x: \ \ \[ x = \frac{1}{3} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural logarithm
A logarithm is a mathematical operation that determines the exponent required to generate a specific number from a given base. When the base is the mathematical constant \(e\) (approximately 2.718), it is called a natural logarithm, and is denoted as \(\ln\). In the equation \(\ln(x) + \ln(3) = 0\), we are using the natural logarithm to solve for \(x\). Natural logarithms are common in calculus and other higher math because \(e\) has unique properties that simplify differentiation and integration.
Combining logarithms
Logarithms possess various properties that facilitate simplification, one of which is the product rule. This rule states:
  • \(\ln(a) + \ln(b) = \ln(ab)\)
Using this property, we can combine the logarithms in our equation \(\ln(x) + \ln(3) = 0\) to get \(\ln(3x) = 0\). By reducing the equation to a single natural logarithm, we simplify the equation, making it easier to solve. Remembering properties like this one will speed up your work with logarithmic equations.
Exponentiation
Exponentiating both sides of an equation can cancel out a natural logarithm, converting a logarithmic expression into a simpler algebraic form. The natural logarithm \(\ln(y)\) is the inverse function of \(e^x\), and thus, \(e^{\ln(y)} = y\). Using this in our equation \(\ln(3x) = 0\), we exponentiate both sides:
  • \(e^{\ln(3x)} = e^0\)
Simplifying this, we find \(3x = 1\). Exponentiation is a powerful tool to isolate logarithmic terms, making equations solvable using basic algebra.
Isolating variables
Solving for a variable typically involves isolating it on one side of the equation. In our final step, we have \(3x = 1\). To isolate \(x\), we divide both sides by 3:
  • \(x = \frac{1}{3}\)
This provides the solution to the equation. Always check your solution by substituting it back into the original equation. Here, \(\ln(\frac{1}{3}) + \ln(3)\) should equal 0, and because \(\ln(\frac{1}{3}*3) = \ln(1) = 0\), our solution is verified.

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Most popular questions from this chapter

Solve each problem. In a certain country the number of people above the poverty level is currently 28 million and growing \(5 \%\) annually. Assuming the population is growing continuously, the population \(P\) (in millions), \(t\) years from now, is determined by the formula \(P=28 e^{0.05 t},\) In how many years will there be 40 million people above the poverty level?

Use a calculator to evaluate each logarithm. Round answers to four decimal places. $$\log (5)$$

Solve each problem. In the diversity index \(d\) for a certain water sample as $$d=\log _{2}\left(\frac{3 \sqrt[3]{2}}{2}\right)$$ Use the base-change formula and a calculator to calculate the value of \(d .\) Round the answer to four decimal places.

Mosquito abatement. In a Minnesota swamp in the springtime the number of mosquitoes per acre appears to grow according to the formula \(N=10^{0.1 t+2},\) where \(t\) is the number of days since the last frost. What is the size of the mosquito population at times \(t=10, t=20,\) and \(t=30 ?\)

Use the base-change formula to find each logarithm to four decimal places. $$\log _{1 / 3}(3.5)$$
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