Math / Algebra

Question9.5-9.6 Warm Up
Write the logarithmic expression as a single logarithm with coeficient 1, and simplify as much as 1) $8 \log _{b} y+6 \log _{b} z$ 2) $6 \log _{8} m-5 \log _{8} n$ 3) $\log _{3} x-4 \log _{3} y-8 \log _{3} z$ 4) $5 \log _{9} m-4 \log _{9} n$ 5) $5 \log _{4} t-4 \log _{4} u-2 \log _{4} v$

Studdy Solution

STEP 1

What is this asking? We need to rewrite each logarithmic expression as a single logarithm and simplify it.
Basically, we're going to combine those separate logs into one super log! Watch out! Remember the properties of logarithms!
We'll be using the power rule and the product and quotient rules.
Don't mix them up!

STEP 2

1. Problem 1
2. Problem 2
3. Problem 3
4. Problem 4
5. Problem 5

STEP 3

We're starting with $8 \log_{b} y + 6 \log_{b} z$ .
Our goal is to combine these two logarithms.

STEP 4

First, let's use the power rule: $n \log_{a} x = \log_{a} x^n$ .
This rule lets us move the coefficients of the logarithms up as exponents.
So, we have $\log_{b} y^8 + \log_{b} z^6$ .
See how the 8 and 6 moved up?

STEP 5

Now, we can use the product rule: $\log_{a} x + \log_{a} y = \log_{a} (x \cdot y)$ .
This lets us combine the sum of two logarithms with the same base into a single logarithm of the product.
Applying this rule gives us $\log_{b}(y^8 \cdot z^6)$ .
Awesome!

STEP 6

We have $6 \log_{8} m - 5 \log_{8} n$ .
Just like before, we'll start with the power rule.

STEP 7

Moving the coefficients up as exponents, we get $\log_{8} m^6 - \log_{8} n^5$ .

STEP 8

This time, we'll use the quotient rule: $\log_{a} x - \log_{a} y = \log_{a} \frac{x}{y}$ .
This lets us combine the difference of two logarithms with the same base into a single logarithm of the quotient.
So, we have $\log_{8} \frac{m^6}{n^5}$ .
Perfect!

STEP 9

We're given $\log_{3} x - 4 \log_{3} y - 8 \log_{3} z$ .
Let's use that power rule again!

STEP 10

Moving the coefficients, we get $\log_{3} x - \log_{3} y^4 - \log_{3} z^8$ .

STEP 11

Applying the quotient rule twice, we have $\log_{3} \frac{x}{y^4} - \log_{3} z^8$ , which simplifies to $\log_{3} \frac{x}{y^4 \cdot z^8}$ .
Fantastic!

STEP 12

We have $5 \log_{9} m - 4 \log_{9} n$ . Power rule time!

STEP 13

Moving those coefficients, we get $\log_{9} m^5 - \log_{9} n^4$ .

STEP 14

Using the quotient rule, we get $\log_{9} \frac{m^5}{n^4}$ .
Wonderful!

STEP 15

We're given $5 \log_{4} t - 4 \log_{4} u - 2 \log_{4} v$ . Power rule, here we come!

STEP 16

Moving the coefficients gives us $\log_{4} t^5 - \log_{4} u^4 - \log_{4} v^2$ .

STEP 17

Applying the quotient rule twice, we have $\log_{4} \frac{t^5}{u^4} - \log_{4} v^2$ , which simplifies to $\log_{4} \frac{t^5}{u^4 \cdot v^2}$ .
Super!

STEP 18

1. $\log_{b}(y^8 \cdot z^6)$
2. $\log_{8} \frac{m^6}{n^5}$
3. $\log_{3} \frac{x}{y^4 \cdot z^8}$
4. $\log_{9} \frac{m^5}{n^4}$
5. $\log_{4} \frac{t^5}{u^4 \cdot v^2}$

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Problem 1
2. Problem 2
3. Problem 3
4. Problem 4
5. Problem 5

STEP 3

We're starting with $8 \log_{b} y + 6 \log_{b} z$ .
Our goal is to combine these two logarithms.

STEP 4

First, let's use the power rule: $n \log_{a} x = \log_{a} x^n$ .
This rule lets us move the coefficients of the logarithms up as exponents.
So, we have $\log_{b} y^8 + \log_{b} z^6$ .
See how the 8 and 6 moved up?

STEP 5

Now, we can use the product rule: $\log_{a} x + \log_{a} y = \log_{a} (x \cdot y)$ .
This lets us combine the sum of two logarithms with the same base into a single logarithm of the product.
Applying this rule gives us $\log_{b}(y^8 \cdot z^6)$ .
Awesome!

STEP 6

We have $6 \log_{8} m - 5 \log_{8} n$ .
Just like before, we'll start with the power rule.

STEP 7

Moving the coefficients up as exponents, we get $\log_{8} m^6 - \log_{8} n^5$ .

STEP 8

This time, we'll use the quotient rule: $\log_{a} x - \log_{a} y = \log_{a} \frac{x}{y}$ .
This lets us combine the difference of two logarithms with the same base into a single logarithm of the quotient.
So, we have $\log_{8} \frac{m^6}{n^5}$ .
Perfect!

STEP 9

We're given $\log_{3} x - 4 \log_{3} y - 8 \log_{3} z$ .
Let's use that power rule again!

STEP 10

Moving the coefficients, we get $\log_{3} x - \log_{3} y^4 - \log_{3} z^8$ .

STEP 11

Applying the quotient rule twice, we have $\log_{3} \frac{x}{y^4} - \log_{3} z^8$ , which simplifies to $\log_{3} \frac{x}{y^4 \cdot z^8}$ .
Fantastic!

STEP 12

We have $5 \log_{9} m - 4 \log_{9} n$ . Power rule time!

STEP 13

Moving those coefficients, we get $\log_{9} m^5 - \log_{9} n^4$ .

STEP 14

Using the quotient rule, we get $\log_{9} \frac{m^5}{n^4}$ .
Wonderful!

STEP 15

We're given $5 \log_{4} t - 4 \log_{4} u - 2 \log_{4} v$ . Power rule, here we come!

STEP 16

Moving the coefficients gives us $\log_{4} t^5 - \log_{4} u^4 - \log_{4} v^2$ .

STEP 17

Applying the quotient rule twice, we have $\log_{4} \frac{t^5}{u^4} - \log_{4} v^2$ , which simplifies to $\log_{4} \frac{t^5}{u^4 \cdot v^2}$ .
Super!

STEP 18

1. $\log_{b}(y^8 \cdot z^6)$
2. $\log_{8} \frac{m^6}{n^5}$
3. $\log_{3} \frac{x}{y^4 \cdot z^8}$
4. $\log_{9} \frac{m^5}{n^4}$
5. $\log_{4} \frac{t^5}{u^4 \cdot v^2}$

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.

Studdy Solution

STEP 1

STEP 2

1. Problem 12. Problem 23. Problem 34. Problem 45. Problem 5

STEP 3

We're starting with 8log⁡by+6log⁡bz8 \log_{b} y + 6 \log_{b} z8logb​y+6logb​z.Our goal is to combine these two logarithms.

STEP 4

First, let's use the **power rule**: nlog⁡ax=log⁡axnn \log_{a} x = \log_{a} x^nnloga​x=loga​xn.This rule lets us move the coefficients of the logarithms up as exponents.So, we have log⁡by8+log⁡bz6\log_{b} y^8 + \log_{b} z^6logb​y8+logb​z6.See how the **8** and **6** moved up?

STEP 5

STEP 6

We have 6log⁡8m−5log⁡8n6 \log_{8} m - 5 \log_{8} n6log8​m−5log8​n.Just like before, we'll start with the **power rule**.

STEP 7

Moving the coefficients up as exponents, we get log⁡8m6−log⁡8n5\log_{8} m^6 - \log_{8} n^5log8​m6−log8​n5.

STEP 8

STEP 9

We're given log⁡3x−4log⁡3y−8log⁡3z\log_{3} x - 4 \log_{3} y - 8 \log_{3} zlog3​x−4log3​y−8log3​z.Let's use that **power rule** again!

STEP 10

Moving the coefficients, we get log⁡3x−log⁡3y4−log⁡3z8\log_{3} x - \log_{3} y^4 - \log_{3} z^8log3​x−log3​y4−log3​z8.

STEP 11

Applying the **quotient rule** twice, we have log⁡3xy4−log⁡3z8\log_{3} \frac{x}{y^4} - \log_{3} z^8log3​y4x​−log3​z8, which simplifies to log⁡3xy4⋅z8\log_{3} \frac{x}{y^4 \cdot z^8}log3​y4⋅z8x​.Fantastic!

STEP 12

We have 5log⁡9m−4log⁡9n5 \log_{9} m - 4 \log_{9} n5log9​m−4log9​n. **Power rule** time!

STEP 13

Moving those coefficients, we get log⁡9m5−log⁡9n4\log_{9} m^5 - \log_{9} n^4log9​m5−log9​n4.

STEP 14

Using the **quotient rule**, we get log⁡9m5n4\log_{9} \frac{m^5}{n^4}log9​n4m5​.Wonderful!

STEP 15

We're given 5log⁡4t−4log⁡4u−2log⁡4v5 \log_{4} t - 4 \log_{4} u - 2 \log_{4} v5log4​t−4log4​u−2log4​v. **Power rule**, here we come!

STEP 16

Moving the coefficients gives us log⁡4t5−log⁡4u4−log⁡4v2\log_{4} t^5 - \log_{4} u^4 - \log_{4} v^2log4​t5−log4​u4−log4​v2.

STEP 17

Applying the **quotient rule** twice, we have log⁡4t5u4−log⁡4v2\log_{4} \frac{t^5}{u^4} - \log_{4} v^2log4​u4t5​−log4​v2, which simplifies to log⁡4t5u4⋅v2\log_{4} \frac{t^5}{u^4 \cdot v^2}log4​u4⋅v2t5​.Super!

STEP 18

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.

1. Problem 1
2. Problem 2
3. Problem 3
4. Problem 4
5. Problem 5

We're starting with $8 \log_{b} y + 6 \log_{b} z$ .
Our goal is to combine these two logarithms.

First, let's use the power rule: $n \log_{a} x = \log_{a} x^n$ .
This rule lets us move the coefficients of the logarithms up as exponents.
So, we have $\log_{b} y^8 + \log_{b} z^6$ .
See how the 8 and 6 moved up?

We have $6 \log_{8} m - 5 \log_{8} n$ .
Just like before, we'll start with the power rule.

Moving the coefficients up as exponents, we get $\log_{8} m^6 - \log_{8} n^5$ .

We're given $\log_{3} x - 4 \log_{3} y - 8 \log_{3} z$ .
Let's use that power rule again!

Moving the coefficients, we get $\log_{3} x - \log_{3} y^4 - \log_{3} z^8$ .

Applying the quotient rule twice, we have $\log_{3} \frac{x}{y^4} - \log_{3} z^8$ , which simplifies to $\log_{3} \frac{x}{y^4 \cdot z^8}$ .
Fantastic!

We have $5 \log_{9} m - 4 \log_{9} n$ . Power rule time!

Moving those coefficients, we get $\log_{9} m^5 - \log_{9} n^4$ .

Using the quotient rule, we get $\log_{9} \frac{m^5}{n^4}$ .
Wonderful!

We're given $5 \log_{4} t - 4 \log_{4} u - 2 \log_{4} v$ . Power rule, here we come!

Moving the coefficients gives us $\log_{4} t^5 - \log_{4} u^4 - \log_{4} v^2$ .

Applying the quotient rule twice, we have $\log_{4} \frac{t^5}{u^4} - \log_{4} v^2$ , which simplifies to $\log_{4} \frac{t^5}{u^4 \cdot v^2}$ .
Super!