For each equation solve one of them by using powers of the same number.to the nearest hundredth, solve the other by using logarithms. 9^x=4

Powers of the Same Number: To solve this equation using powers of the same number, we can express both sides with the same base. Since 9 = 3^2, we can rewrite the equation as: [ (32)x = 4 ] Applying the power of a power rule, we multiply the exponents: [ 3^{2x} = 4 ] Now, let’s solve for (x): [ 2x = \log_3(4) ] [ x = \frac{\log_3(4)}{2} ] Calculating to the nearest hundredth: [ x \approx 0.63 ]

Using Logarithms: We can also solve the equation by taking the logarithm of both sides. Let’s use the natural logarithm (base (e)): [ \ln(9^x) = \ln(4) ] Applying the power rule for logarithms: [ x \ln(9) = \ln(4) ] Solving for (x): [ x = \frac{\ln(4)}{\ln(9)} ] Calculating to the nearest hundredth: [ x \approx 0.63 ]

Therefore, the solution to the equation 9^x = 4 is approximately (x \approx 0.63).

msm555 msm555 · Answer 2 · 2024-03-01T00:10:01+01:00

Answer:

x = 0.63

Step-by-step explanation:

Let's solve the equation [tex]9^x = 4[/tex] using two different methods:

Method 1: Using Powers

To solve [tex]9^x = 4[/tex] using powers of the same number, we can express both sides using a common base. In this case, we can rewrite 4 as [tex]2^2[/tex] because 2 is a power of 9:

[tex] 9^x = 2^2 [/tex]

Now, we can equate the exponents:

[tex] 3^{2x} = 2^2 [/tex]

[tex] 3^{2x} = 4 [/tex]

Now, take the logarithm base 3 of both sides:

[tex] \log_3(3^{2x}) = \log_3(4) [/tex]

[tex] 2x = \log_3(4) [/tex]

[tex] x = \dfrac{\log_3(4)}{2} [/tex]

[tex] x = \dfrac{ \log(4)}{2\log (3)}[/tex]

[tex] x \approx 0.6309297536[/tex]

[tex] x = 0.63 \textsf{ (in nearest hundredth)}[/tex]

Method 2: Using Logarithms

The given equation is [tex]9^x = 4[/tex]. To solve for [tex]x[/tex] using logarithms, take the logarithm base 9 of both sides:

[tex] \log_9(9^x) = \log_9(4) [/tex]

[tex] x \log_9(9) = \log_9(4) [/tex]

[tex] x = \dfrac{\log_9(4)}{\log_9(9)} [/tex]

Now, we know that [tex]\log_9(9) = 1[/tex], so the expression simplifies to:

[tex] x = \log_9(4) [/tex]

[tex] x = \dfrac{ \log(4)}{\log (9)}[/tex]

[tex] x \approx 0.6309297536[/tex]

[tex] x = 0.63 \textsf{ (in nearest hundredth)}[/tex]

So, the value of x is 0.63.