Problem 37
Question
Solve each exponential equation . Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 7^{x+2}=410 $$
Step-by-Step Solution
Verified Answer
The solution of the equation 7^{x+2} = 410 in terms of natural logarithms is x = [ln(410)/ln(7)] - 2. The decimal approximation, correct to two decimal places, for the solution is x ≈ 2.92.
1Step 1: Isolate the Exponential Function
Start by isolating the exponential function 7^{x+2} = 410. It is already isolated in the problem provided.
2Step 2: Apply Logarithm on Both Sides
To get rid of the exponent on one side, apply logarithm. In this case, applying natural logarithm (ln) to both sides gives us: ln(7^{x+2}) = ln(410).
3Step 3: Simplify the Equation by Using Logarithmic Properties
Using the property of logarithms that allows bringing down the exponent as a multiplier, our equation becomes: (x+2)ln(7) = ln(410).
4Step 4: Solve for x
Solving the equation for x will provide the solution in terms of natural logarithms: x = [ln(410)/ln(7)] - 2.
5Step 5: Calculate Decimal Approximation
Using a calculator or a computing tool to compute the value of the right-hand side will provide a decimal approximation of the solution, correct to two decimal places: x ≈ 2.92. Remember the logarithmic base e (Euler's Number) for the natural log in the calculator.
Key Concepts
Natural LogarithmsCommon LogarithmsLogarithmic PropertiesDecimal Approximation
Natural Logarithms
Natural logarithms, denoted as 'ln', are a mathematical concept used to solve equations involving exponential functions. The base of a natural logarithm is the mathematical constant 'e', approximately equal to 2.71828. When you take the natural logarithm of a number, you're asking, 'To what power must e be raised, to produce this number?'.
For example, if we have the equation \( e^y = x \), taking the natural logarithm of both sides (ln) would provide the solution \( y = ln(x) \). It's an essential tool for solving exponential equations because you can effectively 'reverse' the exponent and bring it down to a position where you can handle it algebraically.
For example, if we have the equation \( e^y = x \), taking the natural logarithm of both sides (ln) would provide the solution \( y = ln(x) \). It's an essential tool for solving exponential equations because you can effectively 'reverse' the exponent and bring it down to a position where you can handle it algebraically.
Common Logarithms
Common logarithms are similar to natural logarithms, but they have a base of 10 instead of 'e'. They are simply expressed as 'log'. So, if you have an equation like \( 10^y = x \), taking the common logarithm of both sides will give you the relationship \( y = log(x) \).
Common logarithms are particularly useful when dealing with exponential equations that involve powers of 10, as they simplify the process of solving these equations. In a practical sense, when you're using a calculator to solve for a common logarithm, you'll often see a 'log' button specifically for this function.
Common logarithms are particularly useful when dealing with exponential equations that involve powers of 10, as they simplify the process of solving these equations. In a practical sense, when you're using a calculator to solve for a common logarithm, you'll often see a 'log' button specifically for this function.
Logarithmic Properties
Logarithmic properties are rules that make working with logarithms more manageable. Two key properties are frequently used: the product rule and the power rule. The product rule states that the logarithm of a product is the sum of the logarithms, that is \( log(ab) = log(a) + log(b) \). The power rule, which we often use to solve exponential equations, says that the logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the base: \( log(a^b) = b \cdot log(a) \).
This last property is particularly useful, as demonstrated in the equation from our exercise \( ln(7^{x+2}) = ln(410) \) where we utilize the power rule to bring down the exponent, simplifying the equation considerably to \( (x+2) \cdot ln(7) = ln(410) \).
This last property is particularly useful, as demonstrated in the equation from our exercise \( ln(7^{x+2}) = ln(410) \) where we utilize the power rule to bring down the exponent, simplifying the equation considerably to \( (x+2) \cdot ln(7) = ln(410) \).
Decimal Approximation
Decimal approximation is a process of finding a decimal number close to the exact solution of an equation, generally carried out with the help of a calculator. While exact solutions, especially involving irrational numbers, can be complex and unwieldy, a decimal approximation gives us a practical number that we can work with and understand better.
In the context of logarithms, after finding the solution using natural or common logarithms, as in our exercise \( x = [ln(410) / ln(7)] - 2 \), we often need to use a calculator to approximate this to a decimal form. Calculators typically have the functionality to compute natural and common logarithms, enabling us to approximate the values to a specified number of decimal places, for instance, \( x ≈ 2.92 \) when rounded to two decimal places.
In the context of logarithms, after finding the solution using natural or common logarithms, as in our exercise \( x = [ln(410) / ln(7)] - 2 \), we often need to use a calculator to approximate this to a decimal form. Calculators typically have the functionality to compute natural and common logarithms, enabling us to approximate the values to a specified number of decimal places, for instance, \( x ≈ 2.92 \) when rounded to two decimal places.
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