What is the solution to the logarithmic equation with different bases?

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In summary: So when you change the base, you also have to take into account the exponent. In summary, to solve the equation $\log_9{(x+1)}+3\log_3{x}=14$, you can change the base of the logarithms to either $\dfrac{1}{2}\log_3(x+1)$ or $6\log_9{x}$, resulting in a 7th degree polynomial that requires technology to solve. The change of base formula can be used to convert logarithms to different bases, and in this case, the exponent of the base also needs to be taken into account.
  • #1
karush
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$\tiny{KAM}$
$\log_9{(x+1)}+3\log_3{x}=14$
ok not sure as to best approach to this
assume change the base 9?
 
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  • #2
change

$\log_9(x+1)$ to $ \dfrac{1}{2}\log_3(x+1)$

or …

$3\log_3{x}$ to $6\log_9{x}$

either way will result in a nasty 7th degree polynomial equation that will require technology to solve
 
  • #3
by w|A $x≈80.8579$

but I didn't know you could change the base like that
 
  • #4
change of base formula …

$\log_b{a} = \dfrac{\log_c{a}}{\log_c{b}}$
 
  • #5
skeeter said:
change of base formula …

$\log_b{a} = \dfrac{\log_c{a}}{\log_c{b}}$

derivation …

let $x = \log_b{a} \implies a = b^x$

$\log_c{a} = \log_c{b^x}$

$\log_c{a} = x \cdot \log_c{b}$

$\dfrac{\log_c{a}}{\log_c{b}} = x = \log_b{a}$
 
  • #6
skeeter said:
change

$\log_9(x+1)$ to $ \dfrac{1}{2}\log_3(x+1)$

or …

$3\log_3{x}$ to $6\log_9{x}$

either way will result in a nasty 7th degree polynomial equation that will require technology to solve
where did the $\dfrac{1}{2}$ come from
 
  • #7
$\log_9(x+1) = \dfrac{\log_3(x+1)}{\log_3{9}} = \dfrac{\log_3(x+1)}{2} = \dfrac{1}{2} \log_3(x+1)$
 
  • #8
skeeter said:
$\log_9(x+1) = \dfrac{\log_3(x+1)}{\log_3{9}} = \dfrac{\log_3(x+1)}{2} = \dfrac{1}{2} \log_3(x+1)$
well that's good to know
 
  • #9
The "1/2" is due to the fact that 3 is the square root (1/2 power) of 9.
 

FAQ: What is the solution to the logarithmic equation with different bases?

1. What are logarithms?

Logarithms are mathematical functions that represent the inverse of exponential functions. They are used to solve equations involving exponents and to compare numbers on a relative scale.

2. What is the base of a logarithm?

The base of a logarithm is the number that is raised to a certain power to produce a given number. For example, in the logarithm 2^3 = 8, 2 is the base.

3. How do you calculate logarithms with different bases?

To calculate a logarithm with a different base, you can use the change of base formula: logb(x) = loga(x) / loga(b), where a is the base of the logarithm and b is the desired base.

4. What is the significance of having 78 logs with different bases?

Having 78 logs with different bases allows for more flexibility in solving equations and comparing numbers. It also allows for a more precise representation of numbers on a logarithmic scale.

5. How are logarithms used in science?

Logarithms are used in various scientific fields, such as chemistry, biology, and physics, to express and analyze data with large ranges of values. They are also used in statistical analysis and in various mathematical models.

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