Math / Algebra

Questiona) $\log _{x} 49=2$

Studdy Solution

STEP 1

What is this asking? Find the value of $x$ that makes the logarithm statement true. Watch out! Remember the definition of a logarithm and don't mix up the base and the argument.

STEP 2

1. Rewrite the logarithm as an exponent.
2. Solve for $x$ .

STEP 3

Remember that $\log_{x} a = b$ means $x^b = a$ .
The base of the logarithm becomes the base of the exponent, the result of the logarithm becomes the exponent, and the thing inside the logarithm is what the whole expression equals.

STEP 4

In our case, $\log_{x} 49 = 2$ means $x^2 = 49$ .
See how the $x$ is the base, the 2 is the exponent, and the 49 is what the expression equals.

STEP 5

To get $x$ by itself, we need to undo that square.
The opposite of squaring something is taking the square root!
So, take the square root of both sides of the equation: $\sqrt{x^2} = \sqrt{49}$ .

STEP 6

This simplifies to $|x| = 7$ , because the square root of $x^2$ is the absolute value of $x$ , and the square root of 49 is 7.

STEP 7

The absolute value of $x$ being 7 means $x$ could be 7 or $-7$ .
So, $x = 7$ or $x = -7$ .

STEP 8

However, the base of a logarithm must be positive.
Therefore, $x = -7$ is not a valid solution.

STEP 9

The solution is $x = 7$ .

Was this helpful?

Questiona) $\log _{x} 49=2$

Studdy Solution

STEP 1

What is this asking? Find the value of $x$ that makes the logarithm statement true. Watch out! Remember the definition of a logarithm and don't mix up the base and the argument.

STEP 2

1. Rewrite the logarithm as an exponent.
2. Solve for $x$ .

STEP 3

Remember that $\log_{x} a = b$ means $x^b = a$ .
The base of the logarithm becomes the base of the exponent, the result of the logarithm becomes the exponent, and the thing inside the logarithm is what the whole expression equals.

STEP 4

In our case, $\log_{x} 49 = 2$ means $x^2 = 49$ .
See how the $x$ is the base, the 2 is the exponent, and the 49 is what the expression equals.

STEP 5

To get $x$ by itself, we need to undo that square.
The opposite of squaring something is taking the square root!
So, take the square root of both sides of the equation: $\sqrt{x^2} = \sqrt{49}$ .

STEP 6

This simplifies to $|x| = 7$ , because the square root of $x^2$ is the absolute value of $x$ , and the square root of 49 is 7.

STEP 7

The absolute value of $x$ being 7 means $x$ could be 7 or $-7$ .
So, $x = 7$ or $x = -7$ .

STEP 8

However, the base of a logarithm must be positive.
Therefore, $x = -7$ is not a valid solution.

STEP 9

The solution is $x = 7$ .

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Questiona) log⁡x49=2\log _{x} 49=2logx​49=2

Studdy Solution

STEP 1

What is this asking? Find the value of xxx that makes the logarithm statement true. Watch out! Remember the definition of a logarithm and don't mix up the base and the argument.

STEP 2

1. Rewrite the logarithm as an exponent.2. Solve for xxx.

STEP 3

Remember that log⁡xa=b\log_{x} a = blogx​a=b means xb=ax^b = axb=a.The **base** of the logarithm becomes the **base** of the exponent, the **result** of the logarithm becomes the **exponent**, and the thing *inside* the logarithm is what the whole expression *equals*.

STEP 4

In our case, log⁡x49=2\log_{x} 49 = 2logx​49=2 means x2=49x^2 = 49x2=49.See how the xxx is the **base**, the **2** is the **exponent**, and the **49** is what the expression equals.

STEP 5

To get xxx by itself, we need to undo that square.The opposite of squaring something is taking the square root!So, take the square root of both sides of the equation: x2=49\sqrt{x^2} = \sqrt{49}x2​=49​.

STEP 6

This simplifies to ∣x∣=7|x| = 7∣x∣=7, because the square root of x2x^2x2 is the *absolute value* of xxx, and the square root of 49 is **7**.

STEP 7

The absolute value of xxx being **7** means xxx could be **7** *or* −7-7−7.So, x=7x = 7x=7 or x=−7x = -7x=−7.

STEP 8

However, the **base of a logarithm must be positive**.Therefore, x=−7x = -7x=−7 is *not* a valid solution.

STEP 9

The solution is x=7x = 7x=7.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Questiona) $\log _{x} 49=2$

What is this asking? Find the value of $x$ that makes the logarithm statement true. Watch out! Remember the definition of a logarithm and don't mix up the base and the argument.

1. Rewrite the logarithm as an exponent.
2. Solve for $x$ .

Remember that $\log_{x} a = b$ means $x^b = a$ .
The base of the logarithm becomes the base of the exponent, the result of the logarithm becomes the exponent, and the thing inside the logarithm is what the whole expression equals.

In our case, $\log_{x} 49 = 2$ means $x^2 = 49$ .
See how the $x$ is the base, the 2 is the exponent, and the 49 is what the expression equals.

To get $x$ by itself, we need to undo that square.
The opposite of squaring something is taking the square root!
So, take the square root of both sides of the equation: $\sqrt{x^2} = \sqrt{49}$ .

This simplifies to $|x| = 7$ , because the square root of $x^2$ is the absolute value of $x$ , and the square root of 49 is 7.

The absolute value of $x$ being 7 means $x$ could be 7 or $-7$ .
So, $x = 7$ or $x = -7$ .

However, the base of a logarithm must be positive.
Therefore, $x = -7$ is not a valid solution.

The solution is $x = 7$ .