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Math  /  Algebra

Question1) Find the roots of each equation a) 2=log⁡(x+25)2=\log (x+25) b) 1−log⁡(w−7)=01-\log (w-7)=0 c) 6−3log⁡(2n)=06-3 \log (2 n)=0

Studdy Solution

STEP 1

What is this asking? We need to find the values of xx, ww, and nn that make each of these logarithmic equations true! Watch out! Remember that the logarithm of a negative number or zero is undefined, so we'll need to check our answers to make sure they don't lead to that situation!

STEP 2

1. Solve for *x*
2. Solve for *w*
3. Solve for *n*

STEP 3

Alright, let's **rewrite** our equation 2=log⁡(x+25)2 = \log(x + 25) in exponential form.
Remember, a logarithm is just an exponent in disguise!
Since we don't see a base written for the logarithm, it's secretly a **base 10**!
So, we're saying "10 to the power of 2 equals x+25x + 25".

STEP 4

Let's write that out: 102=x+2510^2 = x + 25.
Now, 10210^2 is just **100**, so we have 100=x+25100 = x + 25.

STEP 5

To **isolate** xx, we'll **subtract 25** from both sides of the equation: 100−25=x100 - 25 = x.

STEP 6

This gives us x=75x = \mathbf{75}!
Let's check: log⁡(75+25)=log⁡(100)=2\log(75 + 25) = \log(100) = 2.
Perfect!

STEP 7

Let's **rearrange** the equation 1−log⁡(w−7)=01 - \log(w - 7) = 0 to isolate the logarithm.
We can **add** log⁡(w−7)\log(w - 7) to both sides to get 1=log⁡(w−7)1 = \log(w - 7).
Again, the base of this logarithm is **10**.

STEP 8

**Rewriting** in exponential form, we have 101=w−710^1 = w - 7, which simplifies to 10=w−710 = w - 7.

STEP 9

Now, we **add 7** to both sides: 10+7=w10 + 7 = w, so w=17w = \mathbf{17}.
Let's check: 1−log⁡(17−7)=1−log⁡(10)=1−1=01 - \log(17 - 7) = 1 - \log(10) = 1 - 1 = 0.
Awesome!

STEP 10

We start with 6−3log⁡(2n)=06 - 3\log(2n) = 0.
Let's **isolate** the logarithm term by **subtracting 6** from both sides: −3log⁡(2n)=−6-3\log(2n) = -6.

STEP 11

Now, we **divide** both sides by −3-3 to get log⁡(2n)=2\log(2n) = 2.
Remember, this is base 10!

STEP 12

**Rewriting** in exponential form gives us 102=2n10^2 = 2n, or 100=2n100 = 2n.

STEP 13

Finally, we **divide** both sides by 2 to find nn: 100/2=n100 / 2 = n, so n=50n = \mathbf{50}.
Let's check: 6−3log⁡(2⋅50)=6−3log⁡(100)=6−3⋅2=6−6=06 - 3\log(2 \cdot 50) = 6 - 3\log(100) = 6 - 3 \cdot 2 = 6 - 6 = 0.
Fantastic!

STEP 14

We found x=75x = 75, w=17w = 17, and n=50n = 50!

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