1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
puteri [66]
2 years ago
10

Log_(5)(x-4)=1-log_(5)(x-8)

Mathematics
2 answers:
hjlf2 years ago
7 0

Answer:

x = 3, x = 9

Step-by-step explanation:

When solving this problem, keep the general format of a logarithm in mind:

b^x=y\\log_b(y)=x

Where, (b) represents the base, (x) is the exponent, and (y) is the evalutaor. Please note that others might use slightly different terminotoly than what is used in this answer.

One is given the following expression, and is asked to solve for the parameter (x);

log_5(x-4)=1-log_5(x-8)

First, manipulate the exquestion such that all of the logarithmic expressions are on one side. Use inverse operations to do this.

(log_5(x-4))+(log_5(x-8))=1

Now use the Logarithmic Base Change rule to simplify. The Logarithmic Base Change rule states the following;

log_b(x)=\frac{log(x)}{log(b)}

Remember, if no base is indicated in a logarithm, then the logarithm's base is (10). Apply the Logarithmic Base Change rule to this problem;

\frac{log(x-4)}{log(5)}+\frac{log(x-8)}{log(5)}=1

Now remove the denominator. Multiply all terms in the equation by the least common denominator; (log(5)) to remove it from the denominator on the left side.

(\frac{log(x-4)}{log(5)}+\frac{log(x-8)}{log(5)}=1)*(log(5))

log(x-4)+log(x-8)=log(5)

All logarithms have the same base, the left side of the equation has the addition of logarithms. This means that one can apply the Logarithm product rule. The logarithm product rules the following;

log_b(x*y)=(log_b(x))+(log_b(y))

This rule can be applied in reverse to simplify the left side of the equation. Rather than rewriting the product of logarithms as two separate logarithms being added, one can rewrite it as one logarithm getting multiplied.

log(x-4)+log(x-8)=log(5)

log((x-4)(x-8))=log(5)

Now used inverse operations to bring all of the terms onto one side of the equation:

log((x-4)(x-8))=log(5)

log((x-4)(x-8))-log(5)=0

Similar to the Logarithm product rule, the Logarithm quotient rule states the following;

log_b(x/y)=(log_b(x))-(log_b(y))

One can apply this rule in reverse here to simplify the logarithms on the left side:

log((x-4)(x-8))-log(5)=0

log(\frac{(x-4)(x-8)}{5})=0

The final step in solving this equation is to use the Logarithm of (1) property. This property states the following:

log_b(1)=0

When applying this property here, one can conclude that the evaluator must be equal to (1), therefore, the following statements can be made.

log(\frac{(x-4)(x-8)}{5})=0

\frac{(x-4)(x-8)}{5}=1

Inverse operations,

\frac{(x-4)(x-8)}{5}=1

(x-4)(x-8)=5

(x-4)(x-8)-5=0

Simplify,

(x-4)(x-8)-5=0

x^2-12x+32-5=0

x^2-12x+27=0

Factor, rewrite the quadratic expression as the product of two linear expressions, such that when the linear expressions are multiplied, the result is the quadratic expression:

x^2-12x+27=0

(x-3)(x-9)=0

Now use the zero product property to solve. The zero product property states that any number times (0) equals (0).

x=3,x=9

Tresset [83]2 years ago
5 0

Hello,

I suppose the question is solve for x.

\displaystyle log_5\ (a)=\dfrac{ln (a)}{ln (5)} \\\\log_5(x-4)=1-log_5(x-8)\\\\\dfrac{ln(x-4)}{ln(5)} =1- \dfrac{ln(x-8)}{ln(5)}\\\\ln(x-4)=ln(5)-ln(x-8)\\\\ln(x-4)+ln(x+8)=ln(5)\\\\ln((x-4)*(x-8))=ln(5)\\\\(x-4)*(x-8)=5\\\\x^2-12x+27=0\\\\\Delta=12^2-4*27=36=6^2\\\\x=9\ or\ x=3\\\\Sol=\{3,9\}\\

For Moderators,

this is a mathematical resolution without any bla-bla sentences that you will easily find. (I can not do it sorry)

You might be interested in
A trapezoid has base lengths of 2.5 inches and 10 inches and an altitude of 8 inches. What is the area of the trapezoid?
lisov135 [29]
A= 50. 2.5+10= 12.5/2= 6.25x8= 50
5 0
3 years ago
Read 2 more answers
Help mehhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
Vanyuwa [196]

Answer:

y = 65x

Step-by-step explanation:

5 0
3 years ago
Read 2 more answers
The graph shows gasoline prices over time. Tell a story about how gas<br> prices change over time.
DochEvi [55]

I'm not sure if this is what you meant, but it was fun to write XD

The market was fairly stable with no problems in the economy and no big political events happening. Then the days started getting colder and people started staying in more- meaning the demand for gas dropped. This eventually it stopped decreasing and stayed like this for a bit until a hurricane is spotted and weather people say it is likely to hit the coast where this gas chart is being charted. The gas prices start increasing, at first its slow as the hurricane gets closer- and it just keeps increasing as people become more and more sure it is going to hit them and they want to evacuate.

7 0
2 years ago
Kristine lives in a region where a proportional income tax rate of 22% applies to everyone. What is the amount of tax she will p
Nitella [24]

Answer:27500

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
Select the correct answer.
scoray [572]

Answer:3

Step-by-step explanation: there is 3 sets of (F), (B), (S)

7 0
3 years ago
Read 2 more answers
Other questions:
  • 1) Sam has 128 marbles . How many hundreds are in this number?
    7·2 answers
  • A/b=c for b<br> Wht is b
    8·1 answer
  • Please help and show work!! Also please check work!!
    7·1 answer
  • A bookstore has a sale: Jen wants to buy four books: for $10.00, for $12.00, for $15.00, and for $20.00. What is the smallest sh
    15·1 answer
  • 6/4 as a whole number or a mixed nimber
    9·1 answer
  • Evaluate f(t)=-2t^2+ 1 for f(-3). PLEASE HELP I BEG YOU!!!!!!!!!!!
    14·1 answer
  • Lyla is planning on riding his bike no more than 65 miles during the week. She has already ridden 17 miles. Write and solve an i
    14·1 answer
  • URGENT 30 POINTS An engineer needs to find the length of a beam to support the beam holding the wall. If the two beams are perpe
    14·1 answer
  • 4<br> Write an exponential decay<br> equation with a rate of 25% and<br> an initial value of 6.
    12·1 answer
  • Use the long division method to find the result when 3xcubed + 16xsquared + 24x + 9 is
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!