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 July 17th, 2012, 09:21 AM #1 Newbie   Joined: Jul 2012 Posts: 18 Thanks: 0 Exponential/logarithmic equations Can I get help with c. 3(4)^3x-2=192 d. 1/3logx=1 e. log2(x+3)+log2(x-3)=4
 July 17th, 2012, 09:42 AM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Exponential/logarithmic equations I have split your post into its own topic. c) $3\cdot4^{3x-2}=192$ Divide through by 3: $4^{3x-2}=64=4^3$ Equate exponents: $3x-2=3$ Now solve for x... d) $\frac{1}{3}\log(x)=1$ Multiply through by 3 $\log(x)=3$ Convert from logarithmic to exponential form: $x=10^3=1000$ e) $\log_2(x+3)+\log_2(x-3)=4$ Observe, we require $3. Use additive property of logs: $\log_2$$(x+3)(x-3)$$=\log_2$$x^2-9$$=4$ Convert from logarithmic to exponential form: $x^2-9=2^4=16$ $x^2=25$ Solve for x, and discard any roots less than or equal to 3.
 July 17th, 2012, 10:08 AM #3 Newbie   Joined: Jul 2012 Posts: 18 Thanks: 0 Re: Exponential/logarithmic equations Thank you so much! Can you also post a clear explanation of why 1/3log2(- has a real solution?
 July 17th, 2012, 10:11 AM #4 Newbie   Joined: Jul 2012 Posts: 18 Thanks: 0 Re: Exponential/logarithmic equations ^im so if it does or does not have a clear solution.
 July 17th, 2012, 10:55 AM #5 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Exponential/logarithmic equations A logarithm of a negative value does not have a real solution, it will be a complex value. This is because (in the case of the given expression) $2^x$ is positive for all real values of x.
 July 17th, 2012, 11:22 AM #6 Newbie   Joined: Jul 2012 Posts: 18 Thanks: 0 Re: Exponential/logarithmic equations Thanks a lot! You really help!
 July 17th, 2012, 11:24 AM #7 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: Exponential/logarithmic equations $\text{Consider thinking about the graph of y= \log_a(x) ...}$ $\text{for all real values of a and x, y is real}$ $\text{Consider the point \log_a(0)= y_0 ...}$ $\text{Implies 0= a^{y_0}}$ $\text{Since the function is undefined at that point, \log_a(0) is a discontinuity. }$ $\text{Now, what is the value of \log_a(x)= y for x \in \mathbb{R}^{-}?}$ $\text{Inverting the function, we get, x= a^y}$ $\text{Now find the domain of a^y for y \in \mathbb{R}}$ $\text{ We see that for y \in \mathbb{R}^{+} \, , \, a^y \in \mathbb{R}^{+}$ $\text{ For y \in \mathbb{R}^{-} \, , \, a^y \in \mathbb{R}^{+}}$ $\text{So, the value of a^y is always positive if y is real.}$ $\text{So we conclude that the value of \log_a(x) for x \in \mathbb{R}^{-} cannot be real}$ $\text{QED}$
 July 18th, 2012, 09:18 PM #8 Newbie   Joined: Jul 2012 Posts: 18 Thanks: 0 Re: Exponential/logarithmic equations cos^2x 1+sinx 1+2sinx-3sin^2x = 1+3sinx solve the trig identity please! with steps shown.
 July 18th, 2012, 09:30 PM #9 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Exponential/logarithmic equations Could you more clearly state the identity you wish to prove? I could venture some guesses, but I think it would be better for you to clearly state the problem.
 July 18th, 2012, 09:39 PM #10 Newbie   Joined: Jul 2012 Posts: 18 Thanks: 0 Re: Exponential/logarithmic equations sorry about that. https://www.desmos.com/calculator there is a link to the equation. Also, 8. Find the sine of the angle formed by two rays that start at the origin of the Cartesian plane if one ray passes through the point (3root3,3) and the other ray passes through the point(-4,4root3) . Round your answer to the nearest hundredth in necessary.

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# The height of a nail in a bicycle tire as it rotates around is periodic in nature. The radius of the tire is 20cm and at the current pedalling speed the tire rotates once every second. Determine the equation of a sinusoidal function that models this situ

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