Solve each exponential equation in Exercises 23–48. 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. 7^0.3x=813

Verified step by step guidance

Start with the given exponential equation: $7^{0.3x} = 813$.

To solve for $x$, take the natural logarithm (or common logarithm) of both sides to utilize the property that $\ln(a^b) = b \ln(a)$: $\ln(7^{0.3x}) = \ln(813)$.

Apply the logarithm power rule to bring down the exponent: $0.3x \cdot \ln(7) = \ln(813)$.

Isolate $x$ by dividing both sides by $0.3 \ln(7)$: $x = \frac{\ln(813)}{0.3 \cdot \ln(7)}$.

Use a calculator to find the decimal values of the logarithms and compute the quotient to get the approximate value of $x$ rounded to two decimal places.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Equations

An exponential equation is one in which the variable appears in the exponent. Solving such equations often involves rewriting the equation to isolate the exponential expression and then applying logarithms to both sides to solve for the variable.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials. They allow us to solve equations where the variable is an exponent by converting the exponential form into a logarithmic form. Common logarithms (base 10) and natural logarithms (base e) are frequently used.

Using Calculators for Approximation

After expressing the solution in logarithmic form, calculators are used to find decimal approximations. This step involves evaluating logarithmic expressions and rounding the result to the desired decimal places, ensuring practical and usable answers.

Recommended video:

5:47

Solving Exponential Equations Using Logs

Related Practice

Textbook Question

Solve each exponential equation in Exercises 23–48. 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

770

views

Textbook Question

In Exercises 39–40, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = log x and g(x) = - log (x+3)

1131

views

Textbook Question

The figure shows the graph of f(x) = e^x. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn h(x) = e^x-1+2

781

views

Textbook Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log$ $\left$( $\frac{10x^2 \sqrt[3]{1 - x}$}{7(x + 1)^2} $\right$)$