Question

$$\text{Solve the following equations.}$$ $$2^x=55$$

Short answer

Question: Solve the equation 2^x = 55 and approximate the value of x. Answer: x ≈ 5.78

Step-by-step solution

Apply a logarithm

In this step, both sides of the equation will be applied with a logarithm (either natural or common logarithm) to eliminate the exponent. Let's use the common logarithm (log): $$\log (2^x)=\log (55)$$

Simplify the equation

Using the logarithm power rule, we bring down the exponent as a coefficient: $$x\cdot \log (2)=\log (55)$$

Solve for x

Now, isolate x by dividing both sides of the equation by the logarithm of 2: $$x=\frac{\log (55)}{\log (2)}$$ Now you can use a calculator to find the numerical value of x: $$x\approx 5.78$$ So the solution to the given equation is: $$x\approx 5.78$$

Key concepts

Exponential Functions in Equations

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. They are written in the form $b^x$, where $b$ is the base and $x$ is the exponent. These functions appear frequently in various fields like biology, finance, and physics because they describe growth and decay processes.

In the context of solving equations, exponential functions require special techniques. If you have an equation like $2^x=55$, you'll need to find the value of $x$ that makes this true. Typically, you'll use logarithms to help bring down the exponent, turning the equation into a solvable form. This is because numeric evaluation alone can be complicated when working directly with exponents.

Importantly, remember these key points about exponential functions in equations:

• The base $b$ is a positive number other than 1.
• They exhibit continuous and smooth behavior bent towards one side.
• Exponential equations can often be transformed into a linear form using logarithms.

Solving exponential equations thus involves turning the problem into something more manageable by utilizing logarithms as a tool.

Understanding the Logarithm Power Rule

The logarithm power rule is a handy mathematical tool used when dealing with exponential equations. It states that the logarithm of a number raised to a power can be rewritten by bringing the exponent out in front as a coefficient. Mathematically, this is expressed as ${\log }_b(a^x)=x\cdot {\log }_b(a)$, where $a$ is the base of the power, $x$ is the exponent, and $b$ is the base of the logarithm.

For instance, in solving the equation $2^x=55$, using the common logarithm (base 10), we apply the logarithm power rule to bring the exponent $x$ down, transforming $\log (2^x)=\log (55)$ into $x\cdot \log (2)=\log (55)$. This simplifies the problem significantly, making it possible to solve for $x$ by straightforward algebraic manipulation.

Here's why this rule is important:

• It simplifies expressions involving exponents, making them easier to handle.
• By transforming the equation, it allows use of simple algebra to find unknowns.
• This approach is particularly useful for equations that can't be solved by simple factorization or arithmetic.

Essentially, the power rule provides a bridge from the complexity of exponents to the simplicity of linear solving.

Practical Use of the Common Logarithm

The common logarithm, often denoted as $\log (x)$, uses 10 as its base. It is an important concept in high school mathematics because it's prevalent in practical calculations and easy to use with calculators. The common logarithm is designed to simplify expressions where precision isn't reliant on the specific base, unlike natural logarithms which are based on $e$. This makes it quite user-friendly for everyday applications.

When solving equations like $2^x=55$, the common logarithm serves as a useful tool to handle the exponent. By converting the equation into logarithmic form $\log (2^x)=\log (55)$, you can use the properties of logarithms such as the power rule to solve for $x$.

Some key points about common logarithms to remember:

• They make complex calculations approachable without needing sophisticated tools.
• Many scientific calculators can readily find common logarithms, simplifying problem-solving.
• Using the common logarithm is helpful in scenarios where the base of the logarithm doesn't need to be specific.

Overall, understanding how to deploy the common logarithm in equations allows for efficient and timely resolution of exponential equations.