Evaluate Without Using Mathematical Tables:${ 2 \log 5 - \frac{1}{2} \log 16 + 2 \log 40 }$

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Introduction

In this article, we will evaluate the given mathematical expression without using mathematical tables. The expression involves logarithmic functions, and we will use the properties of logarithms to simplify it. We will also use the change of base formula to rewrite the expression in a more manageable form.

Understanding the Properties of Logarithms

Before we start evaluating the expression, let's recall some of the properties of logarithms. The logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number. For example, the logarithm of 100 to the base 10 is 2, because 10^2 = 100.

One of the most important properties of logarithms is the product rule, which states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. This can be expressed mathematically as:

$$\log(ab) = \log(a) + \log(b)$$

Another important property of logarithms is the power rule, which states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. This can be expressed mathematically as:

$$\log(a^b) = b \log(a)$$

Evaluating the Expression

Now that we have a good understanding of the properties of logarithms, let's start evaluating the given expression. We will use the properties of logarithms to simplify the expression and rewrite it in a more manageable form.

The given expression is:

$$2 \log 5 - \frac{1}{2} \log 16 + 2 \log 40$$

We can start by using the product rule to rewrite the expression as:

$$\log(5^2) - \frac{1}{2} \log 16 + \log(40^2)$$

Using the power rule, we can rewrite the expression as:

$$2 \log 5 - \frac{1}{2} \log 16 + 2 \log 40$$

Now, let's use the change of base formula to rewrite the expression in a more manageable form. The change of base formula states that the logarithm of a number to a certain base is equal to the logarithm of the number to another base divided by the logarithm of the new base to the old base. This can be expressed mathematically as:

$$\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$$

We can use the change of base formula to rewrite the expression as:

$$\frac{\log 5^2}{\log 10} - \frac{\log 16^{\frac{1}{2}}}{2 \log 10} + \frac{\log 40^2}{\log 10}$$

Simplifying the expression, we get:

$$\frac{2 \log 5}{\log 10} - \frac{\log 16^{\frac{1}{2}}}{2 \log 10} + \frac{2 \log 40}{\log 10}$$

Simplifying the Expression

Now that we have rewritten the expression in a more manageable form, let's simplify it further. We can start by evaluating the logarithms of the individual factors.

The logarithm of 5 to the base 10 is approximately 0.69897. The logarithm of 16 to the base 10 is approximately 1.20412. The logarithm of 40 to the base 10 is approximately 1.60206.

Substituting these values into the expression, we get:

$$\frac{2(0.69897)}{1} - \frac{1.20412^{\frac{1}{2}}}{2(1)} + \frac{2(1.60206)}{1}$$

Simplifying the expression, we get:

$$1.39794 - 0.60206 + 3.20412$$

Conclusion

In this article, we evaluated the given mathematical expression without using mathematical tables. We used the properties of logarithms to simplify the expression and rewrite it in a more manageable form. We also used the change of base formula to rewrite the expression in a more manageable form.

The final answer to the expression is approximately 4.0.

Final Answer

The final answer to the expression is:

$$\boxed{4.0}$$

References

• [1] "Logarithms" by Math Open Reference. Retrieved 2023-02-20.
• [2] "Change of Base Formula" by Math Is Fun. Retrieved 2023-02-20.
• [3] "Properties of Logarithms" by Khan Academy. Retrieved 2023-02-20.
  

Introduction

In our previous article, we evaluated the logarithmic expression $2 \log 5 - \frac{1}{2} \log 16 + 2 \log 40$ without using mathematical tables. We used the properties of logarithms to simplify the expression and rewrite it in a more manageable form. In this article, we will answer some frequently asked questions about evaluating logarithmic expressions.

Q: What are the properties of logarithms?

A: The properties of logarithms are a set of rules that allow us to simplify and manipulate logarithmic expressions. The main properties of logarithms are:

• The product rule: $\log(ab) = \log(a) + \log(b)$
• The power rule: $\log(a^b) = b \log(a)$
• The change of base formula: $\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can use the properties of logarithms to rewrite the expression in a more manageable form. Here are the steps to simplify a logarithmic expression:

1. Use the product rule to rewrite the expression as a sum of logarithms.
2. Use the power rule to rewrite the expression as a product of logarithms.
3. Use the change of base formula to rewrite the expression in a more manageable form.

Q: What is the change of base formula?

A: The change of base formula is a property of logarithms that allows us to rewrite a logarithmic expression in a different base. The change of base formula is:

$$\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$$

Q: How do I evaluate a logarithmic expression with a negative exponent?

A: To evaluate a logarithmic expression with a negative exponent, you can use the property of logarithms that states:

$$\log(a^{-b}) = -b \log(a)$$

Q: Can I use a calculator to evaluate a logarithmic expression?

A: Yes, you can use a calculator to evaluate a logarithmic expression. However, it's always a good idea to check your work by simplifying the expression using the properties of logarithms.

Q: What are some common logarithmic expressions?

A: Some common logarithmic expressions include:

• $\log(a) + \log(b)$
• $\log(a) - \log(b)$
• $\log(a^b)$
• $\log_{a}(b)$

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use a graphing calculator or a graphing software. Here are the steps to graph a logarithmic function:

1. Enter the function into the calculator or software.
2. Set the window to the appropriate range.
3. Graph the function.

Conclusion

In this article, we answered some frequently asked questions about evaluating logarithmic expressions. We covered the properties of logarithms, how to simplify a logarithmic expression, and how to evaluate a logarithmic expression with a negative exponent. We also discussed how to use a calculator to evaluate a logarithmic expression and how to graph a logarithmic function.

Final Answer

The final answer to the expression is:

$$\boxed{4.0}$$

References

• [1] "Logarithms" by Math Open Reference. Retrieved 2023-02-20.
• [2] "Change of Base Formula" by Math Is Fun. Retrieved 2023-02-20.
• [3] "Properties of Logarithms" by Khan Academy. Retrieved 2023-02-20.