Math Problem Statement
Rewrite the equation log_4(x) + log_4(36) = 1 without logarithms.
Solution
ToThe given equation is:
To rewrite the equation without logarithms, we can the use given the logarith equationmic property that states:
(\ \log_blog_ a +4 \log_b{x c} = + \log \_b (alog \cd_ot c) 4{36
Thus, applying this property to the} given equation:
\ **] Now, to eliminate thewithout logarithm, we can rewrite the logarith equation in exponential form.ms Recall that \**,(\log_b a follow = c\ these) steps is equivalent to \(b: ^c =1 a.\ Combine). the Applying this to our equation: \[ 4^1 = logarith x \cdmicot 36 terms$$ Simpl using theifying **: $$4 = x \cdproduct ruleot 36 **\ for] logarithThis is thems equation rewritten: \ without logarithms. [ Would you like further \log_b{A} + \ explanationlog_b or{B} clarification = \ onlog any_b step? Let me know if you'd like to proceed with solving for $$x$$. Here are some related questions you might find interesting{: 1.(A \cdot B)}. How can \ logarithmic equations] be simplified Therefore when: the logarith \ms[ have different bases? 2. What is the exponential form \ of a logarithmic equation? log3._ How do4 you{x solve for $$x$$ in logarithmic equations? 4. What are the basic logarithmic identities that can help in solving equations? 5}. + \ How does changing thelog base of a logarithm affect its properties? **Tip_:**4{ Always remember to use logarithmic identities to combine terms before attempting to eliminate the logarithms.36} = \log_4{(x \cdot 36)}.$$ 2. Rewrite the equation with the single logarithm: $$\log_4{(36x)} = 1.$$ 3. Eliminate the logarithm by converting to exponential form. Recall that $$\log_b{A} = C$$ is equivalent to $$A = b^C$$, where $$b$$ is the base of the logarithm. Here, $$b = 4$$ and $$C = 1$$. So: $$36x = 4^1.$$ 4. Simplify the exponential: $$36x = 4.$$ Thus, the rewritten equation without logarithms is: $$36x = 4.$$ --- Let me know if you'd like the full solution for $$x$$ or further clarifications. Here are some related questions you might find helpful: 1. How do you solve $$\log_b{x} = c$$ for $$x$$? 2. What are the properties of logarithms, such as the product and power rules? 3. How do you rewrite logarithmic equations in exponential form? 4. If $$\log_4{x} + \log_4{y} = z$$, what steps do you take to solve for $$x$$ or $$y$$? 5. What happens when the base of the logarithm changes (e.g., $$\log_4$$ to $$\log_{10}$$)? **Tip:** Always apply logarithmic rules (product, quotient, and power) to simplify expressions before solving.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Logarithmic Equations
Logarithmic Properties
Exponential Form
Formulas
log_b(A) + log_b(B) = log_b(A * B)
log_b(A) = C is equivalent to A = b^C
Theorems
Product Rule of Logarithms
Logarithmic to Exponential Conversion
Suitable Grade Level
Grades 9-11