Question

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.$\ln \sqrt{x+2}=1$

Short answer

The approximate solution for 'x' is 5.389.

Step-by-step solution

Simplify the radical expression

First, note that $\sqrt{x+2}$ can be rewritten in exponential form as $(x+2{)}^{1/2}$. So our equation becomes $\ln (x+2{)}^{1/2}=1$.

Convert logarithmic form to exponential form

The equation $\ln (x+2{)}^{1/2}=1$ can be rewritten in exponential form using the property that if $\ln b=a$, then $b=e^a$. So our equation becomes $(x+2{)}^{1/2}=e^1$.

Solve for 'x'

Square both sides to eliminate the square root on the left side: $(x+2)=(e^1{)}^2$ which simplifies to $x+2=e^2$. Then, subtract 2 from both sides to solve for 'x': $x=e^2-2$.

Approximate the result

Use a calculator to compute $e^2-2$ and round to three decimal places: $x\approx 5.389$.

Key concepts

Solving Logarithmic Equations

To solve logarithmic equations like $\ln \sqrt{x+2}=1$, one must understand the relationship between logarithms and exponents. A logarithm is essentially an exponent itself; it represents the power to which a base number must be raised to produce a given number. Here's a step-by-step breakdown on how to solve the given logarithmic equation:

First, translate the radical expression into exponential form, which simplifies the process. Knowing that the square root is the same as raising to the power of $1/2$, we restate the equation as $\ln (x+2{)}^{1/2}=1$. This simplifies the radical and makes it easier to manipulate algebraically.

Next, by using the conversion between logarithmic and exponential form, we can now rewrite the equation as $e^1$, since the natural logarithm $\ln$ is implicitly with base $e$, the Euler's number. This is a critical step because it converts a logarithm to an exponent, which then allows for straightforward algebraic operations. After this, we solve for $x$ by squaring both sides, which removes the fractional exponent, and then isolating $x$.

Finally, with the value of $e$ known, a calculator can be used to approximate the result to three decimal places. For students, practicing these steps on a variety of problems deepens understanding and hones the skill of manipulating and solving logarithmic equations.

Converting Logarithms to Exponential Form

The ability to convert logarithms to exponential form is crucial for solving logarithmic equations. The conversion leverages the fact that a logarithm answers the question: 'To what power must we raise the base to get a certain number?' Mathematically, if ${\log }_{b}a=c$, then $b^c=a$.

This property underscores the intimate relationship between logarithms and exponents; they are essentially inverse operations. For the natural logarithm $\ln a=b$, we convert to exponential form by raising Euler's number $e$ to the power of $b$, making $e^b=a$.

Understanding this concept and translating between these two forms enable students to easily handle more complex equations involving logarithms. It's a transformation that often simplifies the problem, moving it from logarithmic terrain into the more familiar territory of polynomial equations.

Properties of Logarithms

Logarithms carry distinctive properties that are particularly useful in solving equations. One fundamental property is the Power Rule, which allows the exponent on the logarithm's argument to be moved to the front as a multiplier, expressed as ${\log }_b(a^c)=c{\log }_b(a)$.

In our example, the natural logarithm's argument is raised by $1/2$, and so it could be brought in front as a multiplier, which is helpful in certain contexts.

Apart from the Power Rule, there is the Product Rule that turns the multiplication of two numbers within a logarithm into the addition of two logarithms, and the Quotient Rule which turns division into subtraction. Each of these properties fundamentally allows the transformation of a logarithmic statement into a form that is easier to compute or compare.

Understanding and utilizing the properties of logarithms not only simplifies complex algebraic expressions but also enables one to grasp the deeper concepts at play when it comes to dealing with logarithmic relationships in mathematics.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are written in the form $f(x)=b^x$, where $b$ is a positive real number not equal to 1. The variable $x$ appears as the exponent, which means that the value of the function grows exponentially with $x$.

These functions are characterized by their rapid growth or decay and are commonly used to model situations where something increases at a constant rate percentage-wise, such as in the cases of population growth, compound interest, and radioactive decay.

The natural exponential function, with the base as Euler's number $e$, is particularly important due to its unique properties and its appearance in various areas of science, finance, and mathematics. It's the function's innate rapid growth characteristic that allows it to describe many natural phenomena so accurately. Recognizing how exponential functions operate and how they can be manipulated mathematically is a key skill for students who are delving into higher-level mathematics and its applications.