Question

Solve (a) \(\ln \left(\mathrm{e}^{x}\right)=5000\) (b) \(\ln \left(\mathrm{e}^{x}+10\right)=5\) (c) \(\frac{2}{3} \ln \left(x^{2}+9\right)=3\) (d) \(\log \left(\frac{x}{2}+1\right)=1.5\)

Short answer

Question: Solve the following logarithmic equations: a) \(\ln(\mathrm{e}^{x}) = 5000\) b) \(\ln(\mathrm{e}^{x} + 10) = 5\) c) \(\frac{2}{3} \ln(x^2 + 9) = 3\) d) \(\log(\frac{x}{2} + 1) = 1.5\) Answer: a) \(x = 5000\) b) \(x = \ln(\mathrm{e}^{5} - 10)\) c) \(x = \pm\sqrt{\mathrm{e}^{\frac{9}{2}} - 9}\) d) \(x = 2(10^{1.5} - 1)\)

Step-by-step solution

Simplify the equation

The given equation is \(\ln(\mathrm{e}^{x}) = 5000\). We can simplify it using the fact that \(\ln(\mathrm{e}^{x}) = x\). So, we get \(x = 5000\). Result: \(x = 5000\) #Problem (b)#

Simplify the equation

The given equation is \(\ln(\mathrm{e}^{x} + 10) = 5\). We cannot directly apply logarithmic properties since the argument has a sum of two terms.

Solve for x

If we let \(y = \mathrm{e}^{x} + 10\), we can rewrite the equation as \(\ln(y) = 5\). To solve for \(y\), we can use the definition of natural logarithm: \(y = \mathrm{e}^{5}\). Then, substitute back into our previous definition of \(y\), \(\mathrm{e}^{x} + 10 = \mathrm{e}^{5}\), and solve for \(x\): \(x = \ln(\mathrm{e}^{5} - 10)\). Result: \(x = \ln(\mathrm{e}^{5} - 10)\) #Problem (c)#

Get rid of the fraction

The given equation is \(\frac{2}{3} \ln(x^2 + 9) = 3\). To eliminate the fraction, multiply both sides by 3: \(2 \ln(x^2 + 9) = 9\).

Eliminate the 2 before the logarithm

To remove the 2, divide both sides by 2: \(\ln(x^2 + 9) = \frac{9}{2}\).

Solve for x

Use the definition of natural logarithm to solve for the variable within the logarithm: \(x^2 + 9 = \mathrm{e}^{\frac{9}{2}}\). Solve for \(x^2\) as follows: \(x^2 = \mathrm{e}^{\frac{9}{2}} - 9\), and finally take the square root of both sides to find the possible values for \(x\): \(x = \pm\sqrt{\mathrm{e}^{\frac{9}{2}} - 9}\). Result: \(x = \pm\sqrt{\mathrm{e}^{\frac{9}{2}} - 9}\) #Problem (d)#

Rewrite the equation using exponential definition

The given equation is \(\log(\frac{x}{2} + 1) = 1.5\). We can rewrite it using the exponential definition (recalling that if \(\log_b{y} = x\), then \(y = b^x\)): \(\frac{x}{2} + 1 = 10^{1.5}\).

Solve for x

To solve for \(x\), rearrange the equation as follows: \(\frac{x}{2} = 10^{1.5} - 1\). Then multiply both sides by 2 to isolate \(x\): \(x = 2(10^{1.5} - 1)\). Result: \(x = 2(10^{1.5} - 1)\)

Key concepts

Natural Logarithm

The natural logarithm, represented by the symbol \ \( \ln \ \), is a type of logarithm that has a constant base \( e \), which is approximately equal to 2.71828. This base, \( e \), is known as Euler's number and is very special in mathematics, especially in calculus and exponential growth models. Natural logarithms are used to undo the effect of exponentials and simplify the expression of complex equations.

Here are some key points to remember about natural logarithms:

• The natural logarithm \( \ln(e^x) \) equals \( x \). This property is because raising \( e \) to any power and then taking the natural logarithm of the result brings you back to the original power.
• Solving equations involving \( \ln \) often involves using the exponential function \( e^x \) to isolate the variable.
• The natural logarithm is frequently used in continuous growth models, such as population growth or compound interest where the rate of change is proportional to the current amount.

Understanding these concepts will make solving equations involving natural logarithms straightforward. They allow us to work with exponential functions effortlessly, unlocking complex mathematical models.

Solving Equations

Solving equations is an essential skill in algebra that involves finding the value of the variable that makes the equation true. In problems with logarithmic functions, breaking down each step methodically often helps in understanding how to isolate and solve for the variable.

Consider these strategies when solving logarithmic equations:

• Identify the type of logarithmic equation: Determine if you can simplify the equation using known properties of logarithms.
• Isolate the logarithmic part: Often, you will need to get the logarithmic term by itself before solving for the variable. For example, rewriting or rearranging parts of the equation can make it easier to manage.
• Utilize definitions and exponential forms: In equations such as \( \ln(y) = 5 \), switch to the exponential form \( y = e^5 \) to solve for \( y \). This aids in transforming the equation into a more solvable format.

By patiently applying these techniques, you can solve more complex logarithmic equations effectively. Each step is crucial to navigating the path from a complex equation to a simple solution.

Properties of Logarithms

Logarithms have intrinsic properties that make them extremely useful in solving equations and simplifying expressions. Understanding these properties is key to working through algebraic problems that involve logarithms.

Some important properties of logarithms include:

• **Product Rule**: \( \log_b(M \cdot N) = \log_b(M) + \log_b(N) \). This property helps break down multiplication inside the logarithm into simpler terms.
• **Quotient Rule**: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \). Use this property to handle division within a logarithmic expression.
• **Power Rule**: \( \log_b(M^p) = p \cdot \log_b(M) \). This is particularly helpful when dealing with exponents within logs.
• **Change of Base Formula**: Sometimes it's essential to convert logs from one base to another using \( \log_b(M) = \frac{\log_k(M)}{\log_k(b)} \).

These properties facilitate the process of rewriting and solving log equations. Being comfortable using these rules allows you to tackle problems with greater ease and accuracy, providing a tactical advantage whether you're simplifying expressions or solving for variables.