Express $3 \log _{10} X - \frac{1}{2} \log _{10} Y + 1$ As A Single Logarithm.

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Introduction

In mathematics, logarithmic expressions are a fundamental concept in algebra and calculus. A logarithmic expression is a mathematical statement that involves the logarithm of a number or an expression. In this article, we will explore how to express a given logarithmic expression as a single logarithm. We will use the properties of logarithms to simplify the expression and rewrite it in a more compact form.

The Properties of Logarithms

Before we dive into the problem, let's review the properties of logarithms. The logarithm of a number is the exponent to which a base number must be raised to produce that number. For example, the logarithm of 100 to the base 10 is 2, because 10^2 = 100. The logarithm of a number can be written as:

log_b(x) = y

where b is the base, x is the number, and y is the exponent.

There are several properties of logarithms that we will use to simplify the expression:

  • Product Property: log_b(xy) = log_b(x) + log_b(y)
  • Quotient Property: log_b(x/y) = log_b(x) - log_b(y)
  • Power Property: log_b(x^y) = y * log_b(x)

Expressing the Logarithmic Expression as a Single Logarithm

Now that we have reviewed the properties of logarithms, let's apply them to the given expression:

3log⁡10x−12log⁡10y+13 \log _{10} x - \frac{1}{2} \log _{10} y + 1

We can start by using the Power Property to rewrite the first term:

3log⁡10x=log⁡10x33 \log _{10} x = \log _{10} x^3

Next, we can use the Product Property to rewrite the second term:

12log⁡10y=log⁡10y12\frac{1}{2} \log _{10} y = \log _{10} y^{\frac{1}{2}}

Now, we can use the Quotient Property to rewrite the third term:

1=log⁡10101 = \log _{10} 10

Substituting these expressions back into the original expression, we get:

log⁡10x3−log⁡10y12+log⁡1010\log _{10} x^3 - \log _{10} y^{\frac{1}{2}} + \log _{10} 10

Using the Quotient Property again, we can rewrite the expression as:

log⁡10x3y12+log⁡1010\log _{10} \frac{x^3}{y^{\frac{1}{2}}} + \log _{10} 10

Finally, using the Product Property, we can rewrite the expression as a single logarithm:

log⁡10(x3y12⋅10)\log _{10} \left( \frac{x^3}{y^{\frac{1}{2}}} \cdot 10 \right)

Simplifying the Expression

The expression can be simplified further by evaluating the product inside the logarithm:

log⁡10(x3y12⋅10)=log⁡10(x3⋅10y12)\log _{10} \left( \frac{x^3}{y^{\frac{1}{2}}} \cdot 10 \right) = \log _{10} \left( \frac{x^3 \cdot 10}{y^{\frac{1}{2}}} \right)

Using the Quotient Property again, we can rewrite the expression as:

log⁡10(x3⋅10y12)=log⁡10(x3y12)+log⁡1010\log _{10} \left( \frac{x^3 \cdot 10}{y^{\frac{1}{2}}} \right) = \log _{10} \left( \frac{x^3}{y^{\frac{1}{2}}} \right) + \log _{10} 10

However, we can simplify this expression further by using the fact that log⁡b(b)=1\log_b(b) = 1:

log⁡10(x3y12)+log⁡1010=log⁡10(x3y12)+1\log _{10} \left( \frac{x^3}{y^{\frac{1}{2}}} \right) + \log _{10} 10 = \log _{10} \left( \frac{x^3}{y^{\frac{1}{2}}} \right) + 1

Conclusion

In this article, we have shown how to express a given logarithmic expression as a single logarithm using the properties of logarithms. We started by applying the Power Property to rewrite the first term, then used the Product Property to rewrite the second term. We then used the Quotient Property to rewrite the third term, and finally used the Product Property again to rewrite the expression as a single logarithm. The resulting expression is a simplified form of the original expression, and it can be used to solve problems involving logarithmic expressions.

Example Problems

Here are some example problems that involve expressing logarithmic expressions as single logarithms:

  • Express 2log⁡10x+3log⁡10y2 \log _{10} x + 3 \log _{10} y as a single logarithm.
  • Express log⁡10x−2log⁡10y+1\log _{10} x - 2 \log _{10} y + 1 as a single logarithm.
  • Express 3log⁡10x+log⁡10y−13 \log _{10} x + \log _{10} y - 1 as a single logarithm.

Tips and Tricks

Here are some tips and tricks for expressing logarithmic expressions as single logarithms:

  • Use the Power Property to rewrite terms with exponents.
  • Use the Product Property to rewrite terms with products.
  • Use the Quotient Property to rewrite terms with quotients.
  • Use the fact that log⁡b(b)=1\log_b(b) = 1 to simplify expressions.

Introduction

In our previous article, we explored how to express a given logarithmic expression as a single logarithm using the properties of logarithms. In this article, we will answer some frequently asked questions about logarithmic expressions and provide additional tips and tricks for simplifying them.

Q: What is the difference between a logarithmic expression and a logarithmic equation?

A: A logarithmic expression is a mathematical statement that involves the logarithm of a number or an expression. A logarithmic equation is a mathematical statement that involves the logarithm of a number or an expression, and is typically used to solve for a variable.

Q: How do I simplify a logarithmic expression with multiple terms?

A: To simplify a logarithmic expression with multiple terms, you can use the properties of logarithms to combine the terms. For example, if you have the expression 2log⁡10x+3log⁡10y2 \log _{10} x + 3 \log _{10} y, you can use the Product Property to rewrite it as log⁡10x2+log⁡10y3\log _{10} x^2 + \log _{10} y^3.

Q: What is the difference between a logarithmic expression and an exponential expression?

A: A logarithmic expression is a mathematical statement that involves the logarithm of a number or an expression. An exponential expression is a mathematical statement that involves the exponentiation of a number or an expression. For example, the expression 232^3 is an exponential expression, while the expression log⁡108\log _{10} 8 is a logarithmic expression.

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 fact that log⁡b(b−x)=−x\log_b(b^{-x}) = -x. For example, if you have the expression log⁡1010−2\log _{10} 10^{-2}, you can rewrite it as −2-2.

Q: What is the difference between a common logarithm and a natural logarithm?

A: A common logarithm is a logarithm with a base of 10, while a natural logarithm is a logarithm with a base of e. For example, the expression log⁡10x\log _{10} x is a common logarithm, while the expression ln⁡x\ln x is a natural logarithm.

Q: How do I simplify a logarithmic expression with a fraction?

A: To simplify a logarithmic expression with a fraction, you can use the fact that log⁡b(xy)=log⁡b(x)−log⁡b(y)\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y). For example, if you have the expression log⁡10xy\log _{10} \frac{x}{y}, you can rewrite it as log⁡10x−log⁡10y\log _{10} x - \log _{10} y.

Q: What is the difference between a logarithmic expression and a trigonometric expression?

A: A logarithmic expression is a mathematical statement that involves the logarithm of a number or an expression. A trigonometric expression is a mathematical statement that involves the trigonometric functions of a number or an expression. For example, the expression sin⁡x\sin x is a trigonometric expression, while the expression log⁡10x\log _{10} x is a logarithmic expression.

Q: How do I simplify a logarithmic expression with a square root?

A: To simplify a logarithmic expression with a square root, you can use the fact that log⁡b(x)=12log⁡b(x)\log_b(\sqrt{x}) = \frac{1}{2} \log_b(x). For example, if you have the expression log⁡10x\log _{10} \sqrt{x}, you can rewrite it as 12log⁡10x\frac{1}{2} \log _{10} x.

Conclusion

In this article, we have answered some frequently asked questions about logarithmic expressions and provided additional tips and tricks for simplifying them. We hope that this guide has been helpful in understanding logarithmic expressions and how to simplify them.

Example Problems

Here are some example problems that involve logarithmic expressions:

  • Simplify the expression 2log⁡10x+3log⁡10y2 \log _{10} x + 3 \log _{10} y.
  • Evaluate the expression log⁡1010−2\log _{10} 10^{-2}.
  • Simplify the expression log⁡10xy\log _{10} \frac{x}{y}.
  • Simplify the expression log⁡10x\log _{10} \sqrt{x}.

Tips and Tricks

Here are some additional tips and tricks for simplifying logarithmic expressions:

  • Use the properties of logarithms to combine terms.
  • Use the fact that log⁡b(b−x)=−x\log_b(b^{-x}) = -x to evaluate expressions with negative exponents.
  • Use the fact that log⁡b(xy)=log⁡b(x)−log⁡b(y)\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y) to simplify expressions with fractions.
  • Use the fact that log⁡b(x)=12log⁡b(x)\log_b(\sqrt{x}) = \frac{1}{2} \log_b(x) to simplify expressions with square roots.

By following these tips and tricks, you can simplify logarithmic expressions and rewrite them in a more compact form.