Problem 56
Question
Solve each logarithmic equation in Exercises \(49-92 .\) Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ \log _{2}(x+50)=5 $$
Step-by-Step Solution
Verified Answer
The exact answer to the problem is \(x = -18\)
1Step 1: Convert Logarithm to Exponential Form
To solve the equation \(\log_2 (x + 50)=5\), it's important to write it in exponential form. Based on the properties of logarithms, \(\log_b a = c\) can be re-written as \(b^c = a\). Therefore, the equation \(\log_2 (x + 50) = 5\) can be rewritten as \(2^5 = x + 50\).
2Step 2: Find the value of x
Now the equation is \(2^5 = x + 50\). Compute the value of \(2^5\), which equals to 32. So the equation becomes \(32 = x + 50\). To solve for \(x\), subtract 50 from both sides of the equation to isolate \(x\), hence \(x = 32 - 50\).
3Step 3: Check the domain
After calculation, the value of \(x\) is \(-18\). However, this logarithmic function is defined only for \(x+50 >0\), which implies \(x>-50\). This means, the solution \(x=-18\) is valid as it falls in the domain of the original logarithmic function.
4Step 4: Write the Solution as Decimal
The final step is to represent the solution as a decimal as requested. Since \(x = -18\) is already a decimal number, no further operations are required.
Key Concepts
Solving LogarithmsConverting Logarithms to Exponential FormLogarithm Domain Restrictions
Solving Logarithms
Understanding how to solve logarithmic equations is key to working through algebra and higher levels of mathematics. A logarithm is basically another way to express exponentiation. The equation \(\log_b(a) = c\) asks for what power \(c\) the base \(b\) must be raised to, to get \(a\).
To solve a logarithmic equation, such as \(\log_2(x+50) = 5\), you can start by identifying the base, which is 2, and set up the equation to find the exponent that would result in \(x + 50\) when 2 is raised to that power. In this case, \(2^5 = x + 50\) is the equation we're looking to solve. The solution then involves simple arithmetic operations to find \(x\). It's important to remember that \(x\) must be a part of the domain of the original logarithm for the solution to be valid.
To solve a logarithmic equation, such as \(\log_2(x+50) = 5\), you can start by identifying the base, which is 2, and set up the equation to find the exponent that would result in \(x + 50\) when 2 is raised to that power. In this case, \(2^5 = x + 50\) is the equation we're looking to solve. The solution then involves simple arithmetic operations to find \(x\). It's important to remember that \(x\) must be a part of the domain of the original logarithm for the solution to be valid.
Converting Logarithms to Exponential Form
Converting logarithms to exponential form is an essential step in solving logarithmic equations. This conversion uses the definition of a logarithm: if \(\log _{b}(a) = c\), then it translates to \(b^c = a\) in exponential form. This conversion makes it easier to solve for the unknown.
For the given problem, we convert \(\log_2(x+50) = 5\) to \(2^5 = x + 50\) by recognizing that the logarithm's base is 2 and the result is the exponent on the right-hand side. When we know that 2 raised to the power of 5 is 32, we can substitute and simplify the equation to find the value of \(x\). Understanding this conversion process allows for solving more complex logarithmic equations that may initially seem challenging.
For the given problem, we convert \(\log_2(x+50) = 5\) to \(2^5 = x + 50\) by recognizing that the logarithm's base is 2 and the result is the exponent on the right-hand side. When we know that 2 raised to the power of 5 is 32, we can substitute and simplify the equation to find the value of \(x\). Understanding this conversion process allows for solving more complex logarithmic equations that may initially seem challenging.
Logarithm Domain Restrictions
When solving logarithmic equations, it's crucial to consider the domain restrictions. The domain of a logarithmic function includes all possible values of \(x\) that make the expression within the logarithm greater than zero, since log only applies to positive numbers.
For our example, \(\log_2(x+50)\), the domain is all values of \(x\) such that \(x+50 > 0\), which simplifies to \(x > -50\). Therefore, any solution that results in an \(x\)-value within this range is valid. In the case of \(x = -18\), it satisfies the domain restriction because \(x + 50 = 32\), which is indeed greater than zero. This awareness of the domain is necessary to avoid including any extraneous solutions that don't hold true for the original logarithmic equation.
For our example, \(\log_2(x+50)\), the domain is all values of \(x\) such that \(x+50 > 0\), which simplifies to \(x > -50\). Therefore, any solution that results in an \(x\)-value within this range is valid. In the case of \(x = -18\), it satisfies the domain restriction because \(x + 50 = 32\), which is indeed greater than zero. This awareness of the domain is necessary to avoid including any extraneous solutions that don't hold true for the original logarithmic equation.
Other exercises in this chapter
Problem 55
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