## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

$x =\dfrac{\ln{3}}{\ln{2}}\approx 1.585$
We let $t=2^{x}$, so the equation becomes: $$t^{2}+t-12=0$$ Factor the quadratic trinomial by looking for factors of $-12$ whose sum is $1$ (the numerical coefficient of the middle term). to obtain: $$(t+4)(t-3)=0$$ Use the Zero Product Property by equating each factor to $0$ to obtain the equations: $$t+4 =0$$ and $$t-3=0$$ Solve each equation to obtain: $$t=-4 \text{ and }t=3$$ Since $t=2^{x}=-4$, then we have either $$2^x=-4\quad \text{ or } \quad 2^x=3$$ Note that $2^{x}=-4$ has no real solutions because when $2$ is raised to any real number, the result will never be $-4$. Hence, we will only consider $2^{x}=3$ To find the value of $x$, we apply $\log$ base $2$ to both sides to obtain: $$\log_{2}2^{x}=\log_{2}3$$ Use the properties $\log_a{a^x}=x$ and the change-of-base formula $\log_a{b}=\frac{\ln{b}}{\ln{a}}$ to solve for $x$: $$x=\log_{2} (3)\\ x=\dfrac{\ln{3}}{\ln{2}}\\ x \approx 1.585$$ Thus, the answer is: $x =\dfrac{\ln{3}}{\ln{2}}\approx 1.585$
{"ft_lang_label":"__label__en","ft_lang_prob":0.7263295,"math_prob":1.0000093,"size":1038,"snap":"2021-43-2021-49","text_gpt3_token_len":400,"char_repetition_ratio":0.1131528,"word_repetition_ratio":0.0,"special_character_ratio":0.41811174,"punctuation_ratio":0.077586204,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000098,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-28T12:24:18Z\",\"WARC-Record-ID\":\"<urn:uuid:f6e5c912-ed52-4eb4-9199-4c68d049c88c>\",\"Content-Length\":\"133409\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:068ec00f-d4c4-4520-9cf7-2d88b2921736>\",\"WARC-Concurrent-To\":\"<urn:uuid:b216da9d-ea19-4e98-a233-057b3b799578>\",\"WARC-IP-Address\":\"52.6.216.181\",\"WARC-Target-URI\":\"https://www.gradesaver.com/textbooks/math/precalculus/precalculus-concepts-through-functions-a-unit-circle-approach-to-trigonometry-3rd-edition/chapter-4-exponential-and-logarithmic-functions-section-4-6-logarithmic-and-exponential-equations-4-6-assess-your-understanding-page-337/57\",\"WARC-Payload-Digest\":\"sha1:OEVG2QV76RZIIP5W4WPQMLOYPCFKQDEM\",\"WARC-Block-Digest\":\"sha1:WFLNJ3TVKS7FNTDJ7VFLX2GR3M5HOD4F\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323588284.71_warc_CC-MAIN-20211028100619-20211028130619-00063.warc.gz\"}"}