Problem 70 - Algebra 2

Question

Solve each equation. $$ \log _{0.1}(x+1)=3 $$

Step-by-Step Solution

Answer

The solution to the equation \(\log _{0.1}(x+1)=3\) is \(x = -0.999\).

Step 1: Rewriting Logarithmic Equation to Exponential Form

A logarithmic equation can be transformed into exponential form using the basic logarithm definition. In this case, the exponential form of \(\log _{0.1}(x+1)=3\) is \(0.1^3 = x+1\).

Step 2: Calculating the Value of the Exponent

Calculate the value of the exponent by raising the base (0.1) to the power of 3, which is \(0.1^3 = 0.001\). This gives the equation \(0.001 = x+1\).

Step 3: Solving for x

Solve for \(x\) by subtracting 1 from both sides of the equation. This gives \(x = 0.001 - 1 = -0.999\).

Key Concepts

Exponential Form

In mathematics, the exponential form is a fundamental way to express equations involving exponents. This form is highly useful when working with logarithmic equations. It allows us to transform a logarithm into a more straightforward equation. For instance, if you have a logarithmic equation like \(\log_b(y) = x\), you can convert it into an exponential form, represented as \(b^x = y\). This transformation is based on the definition of a logarithm.

Because the logarithm \(\log_b(y)\) shows the power to which the base \(b\) must be raised to obtain \(y\), converting to exponential form can simplify complex problems. This form changes a logarithm's problem into a more direct arithmetic problem. For instance, in the exercise given, \(\log _{0.1}(x+1)=3\) translates into the exponential form of \(0.1^3 = x+1\).

By using the exponential form we find values more directly. Remember, whenever you're stuck on a logarithmic problem, consider converting it into its exponential form for an easier computation.

Base of Logarithm

A logarithm's base is a crucial component in understanding logarithmic equations. The base tells us the number that is raised to a power. To solve logarithmic equations effectively, you must understand the role of the base. In the equation \(\log_b(y) = x\), \(b\) is the base. It determines the growth factor or the multiplier.

• If the base \(b\) is greater than 1, the function is a growing function.
• If it is between 0 and 1, like 0.1 in this exercise, the function is a decreasing one.

The base must always be positive and cannot equal one because logarithms are undefined for negative or zero bases.Understanding and selecting the correct base is vital. It dramatically influences the calculation results. In our exercise, \(\log_{0.1}(x+1) = 3\), the base 0.1 indicates we are dealing with a decreasing exponential function. It transforms the logarithmic equation to the exponential equation \(0.1^3 = x + 1\). This fundamental concept forms the basis for accurate computations in logarithmic and exponential equations.

Solving Equations

Solving equations, particularly logarithmic ones, requires transforming and manipulating equations to isolate the variable. In the context of a logarithmic problem, you can often make the process easier by converting the logarithm into its exponential form first.After converting your logarithmic equation into exponential form, solving for the unknown variable becomes a simpler task. For example, from \(\log_{0.1}(x+1) = 3\) to the exponential form \(0.1^3 = x + 1\), we can solve for \(x\) by straightforward arithmetic. Calculate \(0.1^3\) to get \(0.001\), and then proceed to solve the equation \(0.001 = x + 1\) for \(x\).

Here, subtracting 1 from both sides gives \(x = 0.001 - 1 = -0.999\).

When solving equations like these, always follow a clear plan:

• Convert to exponential form if necessary.
• Calculate known values or simplify the equation.
• Isolate and solve for the variable.

With these steps, solving equations becomes a methodical and systematic process, paving the way for accurate results.