Solving Geometric Progression & Logarithmic Equations Math Questions

In summary: Solve the equation log_2 x + 2 log_2 3 = log_2(x+5).The equation is log_2 x + log_2 3 = log_2(x+5). To solve for x, use the power rule: log_2 x + log_2 3^2 = log_2(x+5). 3) Find:For #1, try substituting in the values for x and y to see if the equation still holds. For #2, use the logarithmic property to solve for x. For #3, use the power rule and the addition rule to solve for x.
  • #1
I'm clever
4
0
I'm stuck on these three maths questions.

1) In a geometric progression, the sum to infinity is four times the first term.

(i) Show that the common ratio is 3
(ii) Given that the third term is 9, find the first term.
(iii) Find the sum of the first twenty terms.


2) Solve the equation log_2 x + 2 log_2 3 = log_2(x + 5).

3) Find:

200
Σ (3n+2)
n=101

for 3) Should I subtract the series cause it doesn't start with 1?
 
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  • #2
Show us what you have tried. (We can't just give answers here.) For #2 you'll need to use the logarithmic property
[tex]\log_b xy = \log_b x + \log_b y[/tex]
... among others.
 
  • #3
eumyang said:
Show us what you have tried. (We can't just give answers here.) For #2 you'll need to use the logarithmic property
[tex]\log_b xy = \log_b x + \log_b y[/tex]
... among others.

[tex]\log_2 x + 2\log_2 3 = \log_2 (x+5)[/tex]

Power rule:

[tex]\log_2 x + \log_2 3^2 = \log_2 (x+5)[/tex]

Addition rule:

[tex]\log_2 (x \times 3x^2) = \log_2 (x+5)[/tex]

[tex]\log_2 3x^2 = \log_2 (x+5)[/tex]
 
  • #4
I'm clever said:
[tex]\log_2 x + 2\log_2 3 = \log_2 (x+5)[/tex]

Power rule:

[tex]\log_2 x + \log_2 3^2 = \log_2 (x+5)[/tex]

Addition rule:

[tex]\log_2 (x \times 3x^2) = \log_2 (x+5)[/tex]

[tex]\log_2 3x^2 = \log_2 (x+5)[/tex]

Is this right?
 
  • #5
I'm clever said:
[tex]\log_2 x + 2\log_2 3 = \log_2 (x+5)[/tex]

Power rule:

[tex]\log_2 x + \log_2 3^2 = \log_2 (x+5)[/tex]

Addition rule:

[tex]\log_2 (x \times 3x^2) = \log_2 (x+5)[/tex]
Where did that second "x" come from? It should be
[tex]\log_2 (x \times 3^2) = \log_2 (x+5)[/tex]
or
[tex]\log_2 (9x) = \log_2 (x+5)[/tex]

After this, use the property:
if logb x = logb y, then x = y
... and solve for x.
 
  • #6
I'm clever said:
I'm stuck on these three maths questions.

1) In a geometric progression, the sum to infinity is four times the first term.

(i) Show that the common ratio is 3
Are you sure you copied the problem right? My understanding is that unless r < 1 the series won't converge. If the sum to infinity is four times the first term, then I'm getting 3/4 as the common ratio, not 3.
 

1. What is a geometric progression (GP)?

A geometric progression is a sequence of numbers where each term is found by multiplying the previous term by a constant number called the common ratio. The general form of a GP is a, ar, ar^2, ar^3, ..., where a is the first term and r is the common ratio.

2. How do I solve a geometric progression equation?

To solve a geometric progression equation, you can use the formula for the sum of a GP, which is S_n = a(1 - r^n)/(1 - r), where S_n is the sum of the first n terms, a is the first term, and r is the common ratio. You can also use the formula for the nth term of a GP, which is a_n = a*r^(n-1), to find a specific term in the sequence.

3. What is a logarithmic equation?

A logarithmic equation is an equation in which the unknown variable appears in the argument of a logarithm. The general form of a logarithmic equation is log(base b)a = c, where a is the value inside the logarithm, b is the base, and c is the result of the logarithm.

4. How do I solve a logarithmic equation?

To solve a logarithmic equation, you can use the properties of logarithms, such as the power rule, product rule, and quotient rule. You can also use the definition of a logarithm to rewrite the equation in exponential form and then solve for the unknown variable.

5. What are some real-life applications of geometric progression and logarithmic equations?

Geometric progression and logarithmic equations are used in many fields, including finance, biology, and computer science. For example, compound interest in finance follows a geometric progression, and population growth in biology can be modeled using a geometric progression. Logarithmic equations are used in signal processing, data compression, and pH calculations in chemistry.

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