Convert $4^w=1428$ To Logarithmic Form And Use The Change Of Base Formula To Solve For $w$. Round The Answer To 3 Decimal Places.Hint: The Change Of Base Formula Is $\log_b A = \frac{\log A}{\log B}$. W = □ W = \square W = □
Introduction
In mathematics, exponential equations and logarithmic equations are two fundamental concepts that are closely related. Exponential equations involve variables in the exponent, while logarithmic equations involve variables as the base of the logarithm. In this article, we will explore how to convert an exponential equation to logarithmic form and use the change of base formula to solve for the variable. We will use the equation $4^w=1428$ as an example and solve for $w$.
Converting Exponential Equations to Logarithmic Form
To convert an exponential equation to logarithmic form, we need to use the definition of logarithms. The logarithm of a number $a$ to the base $b$ is the exponent to which $b$ must be raised to produce $a$. In other words, if $b^x=a$, then $x=\log_b a$. Using this definition, we can rewrite the exponential equation $4^w=1428$ as a logarithmic equation.
This is the logarithmic form of the original exponential equation. We can see that the base of the logarithm is $4$, and the argument of the logarithm is $1428$.
The Change of Base Formula
The change of base formula is a fundamental concept in logarithms that allows us to change the base of a logarithm from one base to another. The change of base formula is given by:
This formula allows us to change the base of a logarithm from $b$ to any other base, such as $10$ or $e$. We can use this formula to change the base of the logarithm in our equation from $4$ to $10$.
Solving for the Variable
Now that we have the logarithmic form of the equation, we can use the change of base formula to solve for the variable $w$. We can use a calculator to find the values of the logarithms.
Now we can substitute these values into the equation.
Therefore, the value of $w$ is approximately $5.414$.
Conclusion
In this article, we have seen how to convert an exponential equation to logarithmic form and use the change of base formula to solve for the variable. We have used the equation $4^w=1428$ as an example and solved for $w$. The change of base formula is a powerful tool that allows us to change the base of a logarithm from one base to another. We can use this formula to solve a wide range of logarithmic equations.
Rounding the Answer to 3 Decimal Places
Finally, we need to round the answer to 3 decimal places. The value of $w$ is approximately $5.414$, so we can round this value to 3 decimal places.
Therefore, the value of $w$ rounded to 3 decimal places is $5.414$.
Final Answer
Q: What is the difference between an exponential equation and a logarithmic equation?
A: An exponential equation involves variables in the exponent, while a logarithmic equation involves variables as the base of the logarithm. For example, the equation $4^w=1428$ is an exponential equation, while the equation $\log_4 1428 = w$ is a logarithmic equation.
Q: How do I convert an exponential equation to logarithmic form?
A: To convert an exponential equation to logarithmic form, you need to use the definition of logarithms. The logarithm of a number $a$ to the base $b$ is the exponent to which $b$ must be raised to produce $a$. In other words, if $b^x=a$, then $x=\log_b a$. Using this definition, you can rewrite the exponential equation as a logarithmic equation.
Q: What is the change of base formula?
A: The change of base formula is a fundamental concept in logarithms that allows you to change the base of a logarithm from one base to another. The change of base formula is given by:
Q: How do I use the change of base formula to solve for the variable?
A: To use the change of base formula to solve for the variable, you need to substitute the values of the logarithms into the equation. For example, if you have the equation $\log_4 1428 = \frac{\log 1428}{\log 4}$, you can use a calculator to find the values of the logarithms and then substitute them into the equation.
Q: Can I use the change of base formula to solve any logarithmic equation?
A: Yes, you can use the change of base formula to solve any logarithmic equation. The change of base formula allows you to change the base of a logarithm from one base to another, which can be useful in solving logarithmic equations.
Q: How do I round the answer to 3 decimal places?
A: To round the answer to 3 decimal places, you need to look at the fourth decimal place and decide whether to round up or down. If the fourth decimal place is 5 or greater, you round up. If the fourth decimal place is less than 5, you round down.
Q: What is the final answer to the equation $4^w=1428$?
A: The final answer to the equation $4^w=1428$ is $w \approx 5.414$.
Q: Can I use a calculator to solve logarithmic equations?
A: Yes, you can use a calculator to solve logarithmic equations. In fact, calculators are often the easiest way to solve logarithmic equations, especially when the values of the logarithms are not easily determined by hand.
Q: Are there any other ways to solve logarithmic equations?
A: Yes, there are other ways to solve logarithmic equations. For example, you can use the properties of logarithms, such as the product rule and the quotient rule, to simplify the equation and solve for the variable. You can also use the change of base formula to change the base of the logarithm and solve for the variable.
Conclusion
In this article, we have seen how to convert an exponential equation to logarithmic form and use the change of base formula to solve for the variable. We have also answered some frequently asked questions about logarithmic equations and provided some additional tips and resources for solving logarithmic equations.